The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms however). A natural example is the category of quasi-coherent sheaves (which has the category of vector bundles as a subcategory). However, when you are talking about algebraic stacks (which are category theoretic stacks fulfilling extra conditions) they only involve isomorphisms. Note that given any stack restricting to isomorphisms gives a stack in groupoids. This is what one does when one considers the algebraic stack of vector bundles: Start with the stack of vector bundles with arbitrary morphisms. This is not an algebraic stack but restricting to isomorphisms gives one.
General stacks (with non-isomorphisms) are used extensively as they encode the idea of descent. This is somewhat orthogonal to algebraic stacks which try to encode the idea of a moduli problem.
Addendum: All morphisms in a descent datum are isomorphisms (this actually
follows and does not have to be assumed). However full descent means
that you can descend objects (a descent datum of objects comes from an object downstairs) but also arbitrary morphisms (a descent datum of morphisms comes from a morphisms downstairs). These two properties together can be formulated as an equivalence of categories between the category of obejcts downstairs and the category of descent data.
Addendum 1: Charles poses an interesting question. One answer can be based on the fact there seems to also be a philosophical difference between general stacks and algebraic stacks. General stacks are based on the idea that we have some objects and relations between them, the morphisms, that can be glued together over some kind of covering. Hence, usually the objects themselves are the things of main interest and the gluing condition is just an extra (though very important) condition on such objects.
Algebraic stacks on the other hand are things that themselves are glued. The relevant idea is that groupoids are a natural generalisation of equivalence relations. (One can more or less arrive at the idea of a groupoid by thinking of bereasoned equivalence relations, elements do not just happen to be equivalent but there are specific, in general several, reasons for them to be equivalent.)
Having said that, one could start with the fact that an algebraic is the stack associated to a smooth algebraic groupoid (i.e., source and target maps are smooth). This gives a candidate generalisation by just looking at smooth algebraic categories instead. However, no natural examples that are not groupoids comes (at least) to my mind. I think the reason might be the above philosophical distinction.
My understanding is that motivic cohomology has the ability to describe integral values of L-functions up to a constant (and specifically values of zeta functions up to a sign). An example would be Milne's paper "motivic cohomology and values of zeta functions" published in compositio in '88. So, the answer to your question in the bold is yes, but I am sorry I don't have a more detailed answer for you.
Best Answer
I will try to answer this question in a way relevant to more than one field, however, to be honest, I'm rather unconventional in the sense that my experience in this area stems from topological and differentiable stacks rather than algebriaic ones. However, from a formal view point, everything is the same.
So let us work in a "background Grothendieck site", which can be topological spaces, differentiable manifolds, or schemes over a fixed base (the first two with the "open cover topology"). Let's call an object in category a "space".
If G is a group object, and X is a space with an action of G, we can take the corase qoutient. However, this is generally not a "nice space" in the senes that the quotient loses a lot of information about the action. In the context of topological spaces, a "nice quotient" would be one that makes the map $X \to X/G$ into a principal G-bundle. However, you need some really nice conditions on the action for this to work in general. E.g., the action needs to be free.
Note, if we consider the projection $X \to X/G$ "coming from the left and the top" and take the pullback, the action is free if and only if the pullback is $G \times X$.
Now, from G acting on X, we can construct the so-called "action groupoid", which has objects X, and arrows $G \times X$, where $(g,x):x \to gx$. This is a groupoid object in spaces, denote it by Act_G(X). Given a space T, we can pretend it's a groupoid object, with all idenity arrows. We can consider Hom(T,Act_G(X)), where the Hom is taken in the 2-category of groupoid objecs, hence, this Hom gives a groupoid, not just a set (the 2-cells are internal natural transormations). The assignment $T \mapsto Hom(T,Act_G(X))$ defines a presheaf of groupoid on spaces. Moreover, there is a canonical morphism $X \to Hom(Blank,Act_G(X))$ of presheaves of groupoids (where X is identified with its representable presheaf). If form a weak 2-pullback by having this morphism "coming from the left and the top", the pullback becomes $G \times X$, one projetion becomes the "source map" and another the "target map". If we say that $Hom(Blank,Act_G(X))$ is our new qoutient, then "the action becomes weakly free".
So far, everything I did was using groupoids. So where to stacks enter the game? Well, $Hom(Blank,Act_G(X))$ is not a very good quotient because if Y is another space, maps from Y to it don't see $Hom(Blank,Act_G(X))$ as "being like a space". E.g. if we are in topological spaces, we can't define maps from Y into it by defining them on the opens of Y in a way that agrees. (For more explanation see my answer to Stacks in the Zariski topology?). What we have to do is "stackify" the presheaf of groupoids $Hom(Blank,Act_G(X))$, (call its stackification X//G). This makes X//G behaves like a spacein the sense that, e.g. in topological spaces, we can defined maps into it by mapping out of opens in a way that agrees. Since stackification preserves finite weak 2-limits, if we form the same pullback diagram but insetad with respect to $X \to X//G$, we still recover the action grouoid and the action is still "weakly free". Morevoer, the projection $X \to X//G$ becomes a G-torosor (principal G-bundle).
So, just using the groupoid, allowed us to keep track of the isotropy data, but not in a way that we get something like a space. For that we need stacks.
If instead of using the groupoid $Act_G(X)$, we used any groupoid object, we can still stackify its associated presheaf of groupoids. The stacks we get in this way are "geometrical", and give rise to topological, differentiable, and Artin stacks respectively.
A final remark. In the comments, it was said that in some sense groupoids are "atlases for stacks". To see this, let's go to manifolds. Given a manifold M described in terms of an atlas, we can construct a Lie groupoid whose objects are the disjoint union of the elements of the atlas, and where we have an arrows from (x,U_a) to (x,U_b) whenever x is in the intersection of these two. This Lie groupoid's associated stack is the same as the manifold M. More generally, given an orbifold described in terms of charts, we can also construct a Lie groupoid with respect to these charts, and its associated stack "represents the orbifold". In general, you can think of Lie groupoids as "generalized atlases" which describe the geometric object which is their associated stack. Of course, just as a manifold can be described by more than one atlas, a differenitbale stack can be described by more than one Lie groupoid.