Such a function exists in Matlab already: expmv, yet not implemented in scipy.
It would accelerate many computations, e.g. note the use of expm in this thread where all is needed is expm(A) @ v.
Useful also in solving linear dynamical systems, i.e. y1 = expmv(tA, y0) instead of y1 = expm(tA) @ y0.

If would be a welcome contribution, I assume I need: implement, add documentation, and test. Anything else?