Do I need to be licensed to care for my ex husband who lives with my son?

Licensed as what?
Are you not just a family member/friend helping someone?
Are you expecting payment for your help?
If so that's something to discuss with ex hubby or your son.

You do not need to be licensed to care for someone. What you will be doing you already have experience with. My concern is medical appts and you helping him get in and out of a car, etc. You don't want to injure your back. Ex should becable to transfer mostly on his own. If son is going to pay, maybe find a Home Health Aide who can physically help ex. You should be enjoying your retirement. Ex should also be doing as much for himself as possible. He can get his Dr to order in home physical theapy and be taught house to do things for himself. Do not enable or disable him. He should becable to make himself a sandwich.

It’s our son together, we are on very good terms, and my son asked if I can get paid to help take some of the burden off of him. I’m retired.

Is it his biological son or your son? I'm only asking because if you are paying him to provide the care, then you probably don't need to be licensed but he most likely would be considered an employ if he's not a blood relative. I would look at labor and elder care laws for your state as these can vary.