To apply the Mayer-Vietoris sequence, you need subspaces whose interiors cover your space (see e.g. Wikipedia, or Hatcher, p. 149). This is not true in your example, because a k-disk in Rn has empty interior for k<n.
You might also enjoy deriving this result from Alexander duality.
Edit: Carsten makes an excellent point, which is that the hard part is to show that the homology of the complement is independent of the embedding. You did say that this is proved in your book, but I wanted to point out that this is quite difficult and arguably surprising.
1) One embedding that satisfies the conditions of the theorem is the Alexander horned sphere, a "wild" embedding of S2 into S3 (this animation is quite nice too). While it's true that the outer component of the complement has the homology of a point, it is very far from being simply connected -- in fact its fundamental group is not finitely generated. (You can find an explicit description of its fundamental group in Hatcher, p. 170-172.)
2) Every knot is an embedding of S1 into S3. The fundamental group of the complement is a strong knot invariant, and is usually much more complicated than just Z. Since H1 of the knot complement is the abelianization of the knot group, the result you are using implies that all knot groups have infinite cyclic abelianization. This is true (it can be seen nicely from the Wirtinger presentation), but it's not obvious.
3) It is important that the ambient space is a sphere (or equivalently Rn). For a simple example where the theorem breaks down, consider embeddings of S1 into a surface Σg of genus g≥2. Taking g=2 for simplicity, we see that there are three topologically inequivalent ways of embedding a circle into Σ2: A) a tiny loop enclosing a disk; B) a loop encircling the waist of the surface and separating it into two components, each of genus 1; and C) a loop going through one of the handles, which does not separate the surface at all.
The homology groups of Σ2 are H0=Z, H1=Z4, and H2=Z. For both A) and B), the complement of S1 has homology groups H0=Z2 because the curve separates, and H1=Z4. However, for C) we have H0=Z because the complement is connected, and H1=Z3 because we have "interrupted" one of the elements [you can see where it went by looking at the Mayer-Vietoris sequence]. Thus we see that the homology of the complement depends essentially on the embedding into the surface Σg, in contrast with the classical case of embedding a circle into the sphere S2.
I must admit, I don't know whether you can just replace excision by Mayer-Vietoris in the usual Eilenberg-Steenrod axioms. But you can do something slightly different:
While the Eilenberg-Steenrod axioms speak of homology of pairs of spaces, you can define a homology theory on spaces also as a sequence of homotopy invariant functors $h_n$ fulfilling a Mayer-Vietoris sequence (as Steven pointed out, you have just some functorial map $h_n(X) \to h_{n-1}(A\cap B)$, which replace as a datum the boundary maps of the long exact sequence of a pair). Then you define the homology of a pair $(X,A)$ as $h_n(X,A) := \ker(h_n(C_AX) \to h_n(pt))$. Here, $C_A X$ stands for the mapping cone of the inclusion $A\hookrightarrow X$. The Mayer-Vietoris sequence (based on the covering $CA \cup X = C_AX$) implies the long exact sequence of the pair. Excision (i.e. $h_n(X,A) \cong h_n(X-B, A-B)$ for $\overline{B} \subset A^\circ$) is a homotopy fact in the sense that the inclusion $C_{A-B}(X-B) \to C_A X$ is a homotopy equivalence. I must admit, I didn't check the last point in detail, but it should be at least true if $B\to A$ is a closed cofibration and in general you might replace the inclusion via a cylinder construction by a closed cofibration.
Thus, you get a different axiomatization of a homology theory using the Mayer-Vietoris sequence [There is also a third axiomatization of a homology theory, using pointed spaces as input and having a sequence for the cone and a suspension isomorphism as axioms]. Of course, if you have already a functor defined on pairs of spaces, you have to check that something like $h_n(X,A) = h_n(C_AX, pt)$ holds.
I want to comment, I know this "absolute" axiomatization primarly from bordism theory, where you can show Mayer-Vietoris quite easily.
Best Answer
I wrote a paper on precisely this question:
A spectral sequence for stratified spaces and configuration spaces of points. Geom. Topol. 21 (2017), no. 4, 2527–2555.