I've made 4-layer FR4 PCBs 0.8mm thick, but I see that some manufacturers has a limit around 0.4mm.
I don't know if that is the thinnest you can get to a normal price.
I suppose you know that PCBs that thin is flimsy, and not very 'rigid'. I would never have made 'large' PCBs in that thickness. You need to balance the amount of copper on each side too, or it may bend when soldering. Since you talk about microvia, I suppose you want to use BGAs or equally dense footprints, that may need the correct flatness to solder properly.