Lets say someone has a condition in which they are required to pee in the sink as it is more difficult for them to urinate sitting down. Let’s say that individual needs to go to the toilet as their wudhu broke between prayers. Are they allowed to wash their face in that same sink even after they urinate again after having broken wudhu during prayers.

There is nothing wrong using the sink again since that’s the only option the person has due to his or her condition.
Make sure to clean the sink anytime he or she uses it