Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$.

So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli space for framed elliptic curves. If the data of a framing can be expressed in algebraic geometry (maybe as some choice of basis for the first etale cohomology), then it ought to be able to be seen as an algebraic stack. However, in this case, it should be covered by schemes $U_i$ etale over $\mathcal{H}$, and I don't really see what these schemes would be.

notan algebraic stack. The framing uses the standard complex topology, it cannot be expressed in algebraic terms. $\endgroup$Formes modulaires et représentations $\ell$-adiques(Séminaire Bourbaki,211968/69, no. 355). If you can find an unburned copy of Conrad's book on the Ramanujan conjecture, that has a rigorous development (which disagrees with Deligne by a sign). $\endgroup$modelof the upper half plane? $\endgroup$