For each $i = 1,2,\cdots,m$, let $A_i \in \mathbb{R}^{m\times m}$ be full rank, and $v_i \in \mathbb{R}^m$. Let us use the notation $[v_i]_{i=1}^m$ for the $m\times m$ matrix created by stacking $v_i$ as column vectors.
Is it possible to compute the following integral in closed form?
$$ \int_0^t \exp\left( \left[ [A_i]^{-1} ( \exp( -A_i s ) - I_m ) v_i \right]_{i=1}^m \right) \, ds \,. $$
If it helps, we know that the matrix $[v_i]_{i=1}^m$ is symmetric positive semidefinite, and the tensor formed by $[A_i]_{i=1}^m = [a_{ijk}]_{i,j,k=1}^m$ is symmetric in the $i,j$ directions.
This integral likely does not have a closed form solution. In the scalar case (i.e. $m=1$), denoting $A_i = a$ and $v_i = v$, the integral can be computed (via Wolfram Alpha) as
$$ \int_0^t \exp( a^{-1}(e^{-at}-1)v ) = \frac{e^{-v/a}}{a} ( \text{Ei}(v/a) - \text{Ei}(e^{-at}v/a) ) \,, $$
where $\text{Ei}(x)=\int_{-\infty}^x t^{-1}e^{t} dt$ is the exponential integral, which is not an elementary function.
Consequently, it's unlikely the higher dimension cases ($m > 1$) will admit a closed form solution.