@Exterior: The van Kampen theorem for a set of base points is dealt with in Topology and Groupoids, T&G, which also deals with covering spaces and orbit spaces from a groupoid viewpoint. A small correction to one part is here.
I was introduced to groupoids in 1965 by Philip Higgins whose downloadable book Categories and Groupoids is a very good account of the area. A proof of the most general van Kampen type theorem for a set of base points is given in this paper. The nice point is also that the proof, as given there by verifying the universal property, is as easy for groupoids as for groups, and also generalises to higher dimensions. Of course if you give the theorem for groupoids, you have to develop some of the algebra of groupoids, to explain the applications. If you want to see some current developments, try arXiv:1207.6404.
My first paper on this was published in 1967, but my books are still, and I am not sure why, the only topology texts in English to give the general theorem.
Writing the book now called T&G led me to ask in the 1960s about the potential use of groupoids in higher homotopy theory, and this led us after years of trying to a new realm.
I should also say that Massey's nice book "Singular homology" takes the cubical approach. The use of cubical sets with connections has brought cubical sets more into current use. I can give references on that if needed.