Let $X$ be a manifold with submanifold $A \subseteq X$. Let $\Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For example, if $X$ is the upper hemisphere of a sphere and $A \cong S^1$ is the equator, then $\Pi_{1}(X,A) \cong S^1$, and for any two objects there is a single morphism.
Question: Is there any theory developed for such groupoids? I'm looking for some kind of van Kampen theorem allowing me to express $\Pi_{1}(X,A)$ in terms of $\Pi_{1}(A)$ and $\Pi_{1}(X \setminus A)$.
Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem