Good question, that.
I think that there'd be restrictions on GenMods and fertility, so perhaps "legal" GenMods are all sterile. I'm not sure about the illegal (and more sentient) ones, and how the biology would even work there. Most hybrids between even closely related species are sterile. Also, if GenMods could breed, the company wouldn't be able to make untold millions off of them.
If you had a particular interest in a character having that ability, we could probably work something out. There's always the exception to every 'rule'.
And of course you can join! I haven't set any restrictions on species used, trusting people aren't going to pick, like, "rhinocerus" or "manatee" (unless for one of the military applications.. hmm..). But yeah, if you can think of a feasible market for your hybrid human - plant/animal/bird (sex slave, servant, living ornament, whatever a rich and corrupt person would spend millions on, at some risk of prosecution) -- go for it.