AFAIK, once you order the oxygen, your aunt's name, doctor's name and contact number will appear on flight information; mechanics will install the tank above her seat, with mask and tubes all set to go, all she has to do is just to take her seat, put on the mask, flight attendant will come and turn on for her. She just leaves every thing upon arrival. Same goes to next connecting flight.
Be sure to notify United if she changes her routing at last minute for whatever reason. It doesn't hurt either to double check with agents while checking in. Can't tell you about the fee though. Hope everything goes smoothly for you and your aunt...