Just to add in the particular case of doing stuff like going from hreps to vreps of a polyhedron, CDDLib can solve the problem exactly and is wrapped (GitHub - JuliaPolyhedra/CDDLib.jl: cdd wrapper module for Julia. cdd is a library for polyhedra manipulation such as double description and Fourier-Motzkin elimination) and can be used via Polyhedra.jl. I just mention it since it’s the only case I know of that you can do easily from Julia.
Convex can use arbitrary numeric types in theory, but I wouldn’t be surprised if something goes wrong with rationals since it’s not tested and they aren’t very robust (eg with SDPs sqrt(2) scalings are used in the MOI codebase). Using Bigfloats is tested on the Convex side though.