Wow thanks for pointing out how obvious it was to you! But I'm not convinced you're correct. I didn't write the Hamiltonian in terms of the 4-potential, in which case I agree a factor of c would be required; I wrote it (and derived it) in terms of the ordinary magnetic vector potential and the scalar potential, just as its written here. It's very easy to check and see that without the factor of c each term has units of energy. With the factor of c, the vector potential term has units of energy over velocity.