You'd probably want to consider the category of [modules](https://en.wikipedia.org/wiki/Module_(mathematics)), since modules are the appropriate generalizations of vector spaces to arbitrary rings. (Though, John only requires [rigs](https://en.wikipedia.org/wiki/Rig_%28mathematics%29), which are strictly more general.) I suspect that endofunctors on this category would let you swap out the underlying ring in the manner you're suggesting.

(But take this with a grain of salt -- I'm throwing around high-powered rules without quite knowing what they mean!)