If the RV is a motorhome, the chassis parts are very likely metric (with the possible exception of a very few things like lug nuts--though even those are likely to be metric these days). The house parts are more likely to be SAE, though #2 Robertson screws far outnumber hex head fasteners, which are mostly found on appliances and such.
I don't think it's generally necessary to collect an enormous supply of tools to begin with. A #2 and Philips screwdriver, an adjustable wrench, a caulking gun, a razor blade scraper and utility knife, a cheap multimeter, and a tire gauge would be a reasonable starting general tool kit. I'd probably add a small air compressor or even a good bicycle tire pump. To that one could add any comparatively specialized tools unique to the specific unit, like a lug wrench or equivalent breaker bar/socket combination to be able to change a tire, whatever wrenches are needed for any tow ball and hitch tightening or adjustment, that sort of thing.
Sometimes I get the feeling that some here carry enough tools and spare parts in their RV to build another RV in the wilderness should the need come up. Particularly if one isn't traveling deep in the wilderness, that seems a bit excessive to me for most situations.