If the Shimura variety is attached to the Shimura data $h:\mathbb S \to G_{/\mathbb R}$,
and if as usual $K$ denotes the centralizer in $G_{/\mathbb R}$ of $h$,
then the automorphic vector bundles are $G(\mathbb R)$-equivariant bundles on $X:= G(\mathbb R)/K(\mathbb R)$ attached to representations of the algebraic
group $K$.
Specifically, if $V$ is the representation (over $\mathbb C$, say), then the associated
vector bundle $\mathcal V$ is given by $(G(\mathbb R)\times V)/K(\mathbb R).$
Note that if $G = GL_2$ and we are in the modular curve case, then $K$ is equal to
$\mathbb C^{\times}$ (thought of as an algebraic group over $\mathbb R$), its complexification is $\mathbb G_m\times \mathbb G_m$, and so there are two integral
parameters describing its irreps (which are just one-dimensional in this case). One of them is the usual $k$ of $\omega^k$; the other can be chosen so as to correspond to twisting by a power of the determinant, which does not change the underlying
line bundle, but changes the equivariant structure (which we don't think about so explicitly in the classical language).
Now $K$ is the Levi in a parabolic $P$ (defined over $\mathbb R$), and there is an open embedding (of complex analytic spaces)
$X := G(\mathbb R)/K(\mathbb R) \hookrightarrow G(\mathbb C)/P(\mathbb C) =: D$.
(If you look at Deligne's Corvalis article and unravel things, you'll see that this is
how the complex structure on $X$ is defined: in the Hodge-theoretic formulation given there, $P$ is the parabolic preserving the Hodge filtration on the Hodge-structure corresponding to the base point of $X$.)
The automorphic vector bundles are naturally defined on the flag variety $D$
(as $\mathcal V := (G(\mathbb C)\times V)/P(\mathbb C)$), and then restricted to $X$.
(So in the modular curve case, $D$ is the projective line, and $\omega^k$ comes from
$\mathcal O(k)$.)
Now the automorphic bundles, being $G$-equivariant, descend to bundles on each
Shimura variety quotient $Sh(X,K_f)$ of $X \times G(\mathbb A_f)$, and have canonical models defined over the reflex field. In the PEL case, one will be able to construct these canonical models using data from the abelian varieties (think about how the abelian varieties give rise to Hodge structures with Mumford--Tate group equal to $G$ in the first place). In general, this result is due to Milne (see his answer).
Added: A colleague pointed out to me, regarding Kevin's initial question,
that it can be understood much more generally. Namely, all automorphic
vector bundles are locally free coherent sheaves, equivariant under the
action of the finite adeles, but not all coherent sheaves necessarily
have these properties! Practically nothing is known about the $K$-theory of
Shimura varieties, although the question is obviously of fundamental
interest.
The very deep current work on cycles on Shimura varieties, from various
points of view, must be the beginning of a substantial theory whose ultimate
shape no one is in a position to imagine. So Kevin's question should be
seen
as a program for future collaboration between number theorists and algebraic
geometers (at least).
Also, let me remark that I'm told that the term "automorphic vector bundle" was invented in
a conversation between Michael Harris and Jim Milne in the second half of the 80s.
One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the Witten genus. The original version of the Witten genus associates power series invariants in $\mathbb{C}[[q]]$ to oriented manifolds, and it was argued that what it calculates on M is an $S^1$-equivariant index of a Dirac operator on the free loop space $Map(S^1,M)$. It is also an elliptic genus, which Ochanine describes much better than I could here.
This is supposed to have especially interesting behavior on certain manifolds. An orientation of a manifold is a lift of the structure of its tangent bundle from the orthogonal group $O(n)$ to the special orthogonal group $SO(n)$, which can be regarded as choosing data that exhibits triviality of the first Stiefel-Whitney class $w_1(M)$. A Spin manifold has its structure group further lifted to $Spin(n)$, trivializing $w_2(M)$. For Spin manifolds, the first Pontrjagin class $p_1(M)$ is canonically twice another class, which we sometimes call "$p_1(M)/2$"; a String manifold has a lift to the String group trivializing this class. Just as the $\hat A$-genus is supposed to take integer values on manifolds with a spin structure, it was argued by Witten that the Witten genus of a String manifold should take values in a certain subring: namely, power series in $\Bbb{Z}[[q]]$ which are modular forms. This is a very particular subring $MF_*$ isomorphic to $\Bbb{Z}[c_4,c_6,\Delta]/(c_4^3 - c_6^2 - 1728\Delta)$.
The development of the universal elliptic cohomology theory ${\cal Ell}$, its refinement at the primes $2$ and $3$ to topoogical modular forms $tmf$, and the so-called sigma orientation were initiated by the desire to prove these results. They produced a factorization of the Witten genus $MString_* \to \Bbb{C}[[q]]$ as follows:
$$
MString_* \to \pi_* tmf \to MF_* \subset \Bbb{C}[[q]]
$$
Moreover, the map $\pi_* tmf \to MF_*$ can be viewed as an edge morphism in a spectral sequence. There are also multiplicative structures in this story: the genus $MString_* \to \pi_* tmf$ preserves something a little stronger than the multiplicative structure, such as certain secondary products of String manifolds and geometric "power" constructions.
What does this refinement give us, purely from the point of view of manifold invariants?
The map $\pi_* tmf \to MF_*$ is a rational isomorphism, but not a surjection. As a result, there are certain values that the Witten genus does not take, just as the $\hat A$--genus of a Spin manifold of dimension congruent to 4 mod 8 must be an even integer (which implies Rokhlin's theorem). Some examples: $c_6$ is not in the image but $2c_6$ is, which forces the Witten genus of 12-dimensional String manifolds to have even integers in their power series expansion; similarly $\Delta$ is not in the image, but $24\Delta$ and $\Delta^{24}$ both are. (The full image takes more work to describe.)
The map $\pi_* tmf \to MF_*$ is also not an injection; there are many torsion classes and classes in odd degrees which are annihilated. These actually provide bordism invariants of String manifolds that aren't actually detected by the Witten genus, but are morally connected in some sense because they can be described cohomologically via universal congruences of elliptic genera. For example, the framed manifolds $S^1$ and $S^3$ are detected, and Mike Hopkins' ICM address that Drew linked to describes how a really surprising range of framed manifolds is detected perfectly by $\pi_* tmf$.
These results could be regarded as "the next version" of the same story for the relationship between the $\hat A$-genus and the Atiyah-Bott-Shapiro orientation for Spin manifolds. They suggest further stages. And the existence, the tools for construction, and the perspective they bring into the subject have been highly influential within homotopy theory, for entirely different reasons.
Hope this provides at least a little motivation.
Best Answer
Here are two applications that I know of which involve direct crossover.
In section 5 of the Hopkins ICM address referenced in Drew's answer to the other question, he gives a topological proof of the following congruence originally due to Borcherds. Suppose that $L$ is a positive definite, even unimodular lattice of dimension $24k$. The theta function $\theta_L$ is the generating function $$ \theta_L(q) = \sum_{\ell \in L} q^{\tfrac{1}{2} \langle \ell, \ell \rangle}, $$ a modular form of weight $12k$ over $\Bbb Z$. If we write $$ \theta_L(q) = c_4^{3k} + x_1 c_4^{3(k-1)} \Delta + \cdots + x_k \Delta^k, $$ in terms of the standard basis elements $c_4$ and $\Delta$, then $x_k \equiv 0$ mod 24.
(However, later in the same paper Hopkins translated the input from topology into a proof in analytic terms.)
There is also the following, due to work of Ando-Hopkins-Rezk and alluded to in the final section of the ICM address. Let's suppose that we fix a prime $p$ and a level for modular forms which is relatively prime to $p$, and for simplicity either that $p > 3$ or that the modular curve is actually a scheme. We have an associated (graded) ring of integral modular forms $MF_*$. Then there is a commutative square: $$ \require{AMScd} \begin{CD} (MF_{> 0})^\wedge_p @>>> (MF_*)^\wedge_{(p,E_{p-1})}\\ @VVV @VVV\\ (E_{p-1}^{-1} MF_*)^\wedge_p @>>> (E_{p-1}^{-1} (MF_*)^\wedge_{E_{p-1}})^\wedge_p \end{CD} $$ Here $E_{p-1}$ is an appropriately normalized Eisenstein series. In degree $k$:
the top map is the Hecke operator $1 − T(p) + p^{k−1}$, which has the Eisenstein series in its kernel;
the left map is the operator $(1 − p^{k−1}V)$ where $V$ is the Verschiebung operator $V(\sum a_n q^n = \sum a_{pn} q^n)$;
the right map is just the localize-then-complete map, which restricts a function near the supersingular locus to the punctured neighborhood of the supersingular locus;
the bottom map is $1 - U_p$ where $U_p$ is the Atkin operator $U_p(f(q)) = f(q^p)$.
What Ando-Hopkins-Rezk showed is that for $k \geq 2$, this square is biCartesian: it becomes a short exact sequence. I'm not aware of a purely number-theoretic proof of this result or its generalizations, though I'd love to hear one.
Having said all this, I don't know of that many results where we know something purely from topology that wasn't already known in number theory. As you say, there are certainly results from number theory applied to topology. Even more common is for the work on the topology side to lead to a question in number theory which is interesting to us but possibly outside mainstream research questions. After all, most of our concerns translate into questions about integral modular forms, operators on them, their congruences, torsion goodies, etc. for a fixed level; rational or analytic information, automorphic representations, etc. have had far less topological impact.
Sometimes topologists prove number-theoretic results as part of this search, but rarely has topological input been required for the proofs.