If we consider mass and charge to be excitations of independent quantum fields, do they necessarily travel together? Can we have objects with only an excitation on the mass field, and objects with only an excitation on the charge field? Could the latter be light?
There are no mass and electric charge fields. Instead, there is an electron field, a photon field, a quark field, a gluon field, etc. Mass and electric charge are properties of field quanta. Electrons and quarks both have mass and electric charge; photons and gluons have neither.
You can definitely have mass without electric charge. For example, a Z boson does, and neutrinos too (since they are believed to have a small mass).
There are no known particles with electric charge but without mass.