While it's true that some Republic of Cypriot companies are running coach trips to the north I'm not aware of any legal arrangement for arriving on one side of the Green Line in order to stay on the other. It's not much of a problem if you live here and cross in your own car (rental cars are still largely confined to their own sides of the border by the hire companies, though it's not actually illegal to cross from South to North and back. It will in all likelihood break the terms of your rental agreement, though. It is forbidden for Northern Cyprus hire cars to enter the South).
Personally, If I were bringing an elderly person with walking difficulties to Cyprus I'd probably look at the hotel district of LImassol; there's a huge range of hotels of all styles and budgets, more restaurants than you could shake a stick at and wheelchair-friendy access to most of the area. There's a flat path/wheelchair route/cycle path right along the beachfront into Limassol (about 4-5 miles in total)