Can an inmate get a divorce free while in prison?

It matters a lot what state you're in. I'll answer it as though you're in Texas. Yes, you can file the Petition by mail to get the divorce started, but it's highly unusual for the judge to sign the final papers without either husband or wife standing before the judge with agreed papers to be signed, and it'd be difficult for you to do that while locked up. The judge won't sign a capias or bench warrant to have you brought before him/her to finalize the divorce.