Here is one way to construct an object $X_{mdR}$ for $X$ say, a complex variety. Recall that $X_{dR}$ can be constructed as follows. Let $\hat{X}$ be the formal completion of the diagonal inside $X\times X$. Then $X_{dR}$ is the quotient of $X$ by the equivalence relation defined by $\hat{X}$.
Let $\eta$ denote the generic point of $X$. We can define an affine scheme $\hat{\eta}'$ as the inverse limit of the $\hat{U},$ as $U$ ranges over all affine opens in $X$. (I believe, though I didn't carefully check, that this is not the formal completion of $\eta$ inside $\eta\times\eta$.) It comes with two natural maps $p_1,p_2:\eta'\rightarrow\eta$ making $\hat{\eta}'$ an equivalence relation. Then we can define $X_{mdR}$ to be the quotient of $\eta$ by $\hat{\eta}'$.
A vector bundle on $X_{mdR}$ is a vector bundle $V$ on $\eta$ with an isomorphism $p_1^*V\cong p_2^*V.$ We can find an open $U_1$ so that $V$ can be defined over $U_1$, and we can find a smaller open $U_2\subseteq U_1$ so that the isomorphism $p_1^*V\cong p_2^*V$ is defined over $\hat{U_2}$. (This follows from the general framework of "Noetherian approximation", but can also be seen explicitly.) Thus $V$ comes from a vector bundle with connection on $U_2$.
That being said, I'm not convinced that this is a good way of thinking about meromorphic connections. My experience is that in non-finite type settings, usually trying to work with $D$-modules via constructions like the de Rham stacks runs into issues fairly quickly. For starters, I'm not sure if this works well in families, i.e., if there is any nice interpretation of vector bundles on $A\times X_{mdR}$.
