Electric charges emit virtual photons that then yield attraction or repulsion of other charges.

Indeed, according to the Maxwell equations, the electric and magnetic fields are proportional to the sourcing charge $q$ (linear in $q$). These fields get into equations of motions of the other charges and determine the corresponding force. This is the way that nature realizes attraction/repulsion of different/like charges.

Does the chirality of the emitted virtual photons depend on the sign of the emitting charge?

If you speak of the angular momentum of the emitted field, it is proportional to the vector product of electric and magnetic fields, so it is quadratic in $q$.