What would the fundamental group of disjoint union look like?

As mentioned in the comments, there is no such thing as the fundamental group of a space. What there is is the fundamental group of a space $X$ at a basepoint $x$, which is "unique up to isomorphism" if $X$ is path-connected. But a nontrivial disjoint union of spaces is never path-connected, so you really need to pick a basepoint. If you pick a basepoint $x \in X_i$, the fundamental group will just be $\pi_1(X_i, x)$.

This is a simple example of what fundamental groupoids can do for you. Unlike the fundamental group, the fundamental groupoid does not require a choice of basepoint. The fundamental groupoid of a disjoint union is just the disjoint union of fundamental groupoids.

This fussing with basepoints may seem like pedantry but it becomes quite important once you want to discuss any extra structure. For example, suppose a group $G$ acts on $X$. Does it act on "the" fundamental group? The answer is not necessarily, not even after choosing a basepoint, because $G$ may not preserve any point of $X$! But $G$ always acts on the fundamental groupoid.