When I thought of using fundamental groupoids in van Kampen type situations in 1964-5, I thought it was great to get rid of base points. But then I realised that to compute the fundamental group of the circle I had to reduce to $2$ base points. So my published paper in 1967 used the fundamental group $\pi_1(X,S)$ on a set $S$ of base points, and this idea was developed in the book published in 1968, and now available as Topology and Groupoids (T&G). This is the only topology text in English to use this notion (why is that?). So it is, as Peter May described the 1988, differently titled, edition, "idiosyncratic". However this paper on ODEs has an application of the notion!
Here is an example of a connected union of 3 non connected spaces:
Note that the fundamental groupoid on the shown set of base points will still capture the symmetry of the situation. Methods of computing colimits of groupoids are analogous to those for groups except that one needs the additional construction of a new groupoid $f_*(G)$ on the set $Y$ from a groupoid $G$ and a function $f:Ob(G) \to Y$. See T&G and Philip Higgins 1971 book Categories and Groupoids (downloadable) (these write $f_*(G)$ as $U_f(G)$). This construction corresponds to identifying some points of a set of base points, i.e. identifications in dimension $0$. This can't be seen with just one base point, and has useful analogues in higher dimensions.
To come nearer to the question, and to Qiaochu's answer, for a disjoint union of connected spaces, you can choose one base point in each component.
So I feel the fundamental groupoid itself is often too big, and the fundamental group is very often too small, but the fundamental groupoid on a set of base points can be chosen to be just right.
To deal with actions of a group on $X$ you need to take $S$ as a union of orbits of the action. This is developed in Chapter 11 of T&G.