I don't know an answer to your question, but there is already a literature on D-modules on algebraic stacks (over $C$ say) and I'm not sure I would expect a significant difference with your topic, though I might be missing something. The basic reference for this is Chapter 7 (essentially a book in itself) of Beilinson and Drinfeld's "Quantization of Hitchin Hamiltonians and Hecke Eigensheaves", available here. There's also a treatment in my paper with David Nadler
on Character Theory of a Complex Lie Group, here, in a more homotopical language, and I would imagine most of the arguments translate to your setting, though perhaps I'm missing something..

Edit: to be slightly more detailed, the argument is explained in Section 4.1 of our paper (I'll try to paraphrase to your setting). You write a smooth stack as a colimit of spaces (say as a geometric realization of a simplicial analytic space) with smooth maps, this defines the constructible derived category (or more precisely its $\infty$-version) as a (homotopy) limit of categories along the simplices with maps given by pullbacks ($f^!$'s in our setting -- those are the natural functors on all $D$-modules). By base change and composition you find that $f_*$ and $f^!$ are then defined for maps of smooth stacks. You also check Verdier duality descends (after normalizing by dimensions) to this limit. That gives you the other two functors, and the adjunction relations descend automatically as well.