Yes, it makes sense if for example your random variables are in $L^1$.

Your map $Z(a)=aX+(1-a)Y$ is a well-defined map from an open set in a Banach space to a Banach space, $Z: R \mapsto L^1$.
In such situation, you can talk about the Frechet derivative of Z, and it satisfies the usual properties you can expect from a derivative.

If you are dealing with random variables living in non-locally convex topological vector spaces (e.g. in $L^p$, $0\leq p < 1$), then I think you run quickly into several problems.

The standard procedure to prove results from calculus for a vector-valued function Z is to go back to a real-valued function just by replacing Z with $\lambda(Z)$, where $\lambda$ is a continuous linear functional on the vector space. That is, we are just looking at the "coordinates" of Z. But if there are no non-zero linear functionals on the vector space (e.g. $L^p$, $0\leq p<1$), then there is not much that can be done.