For people that need wheelchairs, what do they do if they need to go to the bathroom on an airplane?

Carl is partly right - the aisle chair is the horrible device he described (it is also used to get the wheelchair user on and off the plane) but as I recollect the rules, it is only REQUIRED on planes over a certain size, or flights over a certain expected duration.  There are also requirements for it to be loaded if a passenger requests it, again with some conditions on plane size and flight length... An additional reason you seldom see it used, is that the flight attendants are only required to offer very limited assistance, and even in the so-called "accessible" plane toilets, it is extremely difficult to transfer or otherwise use the facility.  It is even worse when you have to use the standard sized toilet... Every article that I've ever seen on advice for the handicapped traveler is to go just before boarding the plane, and endeavor to limit your fluid intake to just what you need to avoid dehydration so that you can wait till the plane lands before needing to go again...  Some articles even suggest possibly having a Foley (indwelling) catheter placed temporarily for extremely long flights. (Obviously individuals that normally use Foley's have less of a concern about this.)

If you are male, and only need to urinate, you can look into wearing a male external catheter like https://www.finneganmedicalsupply.com/detail/Coloplast-Conveen-Optima-Male-External-Catheters that connects with tubing to a leg bag like https://www.finneganmedicalsupply.com/detail/Coloplast-Conveen-Security-Contoured-Leg-Bag.  There is a limited volume of collection, so you would still need to restrict fluids on longer flights.  I would also recommend using the toilet prior to boarding or requesting a seat near the toilet when booking your ticket. If you are currently used to wearing a Foley, or indwelling, catheter, that is certainly an option but it would still need to be attached to a drainage bag of some sort.  Also I would not necessarily recommend a Foley unless you regularly use one.  I say this because indwelling catheters increase the risk of UTIs.

Every flight I have been on I have used the chair above. It's quite handy, but only if you are the right size. I'm very small, about 100 pounds and easily fit, but I could see how it would be a problem from anyone much bigger. Usually I try to wait until I'm off the plane, but on flights longer than 6 hours it can be difficult.

We don’t. Think of that when you are on the plane next time. This is a very difficult thing for us but no one cares to address it.