Any vector bundle is a groupoid: you can add and subtract vectors only if they are in the same fibre. Similarly, if you take a vector bundle E → M (or some other fibre bundle) then consider the automorphism bundle Aut(E) → M where a point in Aut(E) above p ∈ M is an automorphism of E_p. This is a groupoid since these automorphisms can only be composed if they lie in the same fibre.

These are discrete groupoids in the sense that there are no morphisms between distinct objects (aka points in the base space). However, they are not discrete topologically as they clearly have topologies! (Which, incidentally, shows that you should be careful when using the statement about groupoids having no homotopy above degree 2: this is a statement about groupoids in Set but groupoids exist enriched in other categories where they can have lots of interesting homotopy). To get a more general groupoid, you can consider the bundle Iso(E) → M × M where a point in Iso(E) above (p,q) ∈ M × M is an isomorphism from E_p to E_q.

One reason for liking groupoids is that they allow you to talk about quotients by group actions without actually having to take the quotient. That's useful because some categories don't have quotients - such as the category of smooth manifolds. So when you have a group G acting on a manifold M you can try to take the quotient M/G but that is quite often not a manifold. So instead you can take the groupoid with objects M and morphisms G × M, where a morphism (g,m) has source m and target gm. Even when the quotient exists, or when you've extended the category to include quotients, this can be much better behaved than the corresponding quotient.