In Ronnie Brown's Topology and groupoid, pg 263, 7.2.1
If $p:E\to B$ is a fibration of groupoids, there is an assignment, $$b_\#:\pi_0 p^{-1}[u]\to\pi_0 p^{-1}[v]$$
I did not see anywhere where he makes a definition of $\pi_0 X$ when $X$ is a groupoid. What is it?
Judging from proof, I am assuming it is the class of objects up to isomoprhism. (Is it even a set?)
Yes, $\pi_0(G)$ should be the class of objects modulo isomorphism.
Note this is also consistent with the interpretation of groupoids as models for homotopy 1-types, since $\pi_0(G) \cong \pi_0(|G|)$ for any groupoid, and $\pi_0(X) \cong \pi_0(\Pi_1 (X))$ for a topological space $X$.