Does the park have any way to pick up handicap people from the Bart station and bring them to the park and then return them to Bart?

Not that I know of but Muni is a great option if you are at Embarcadero.

Uber?

Yes they have a separate disabled/elderly entrance with an elevator

Muni is likely the best option. But, it can get crowded, especially after the last out. You could wait it out. But, I wouldn't recommend it after a night game. If you need to get to Alameda, Oakland, or Vallejo, there's the ferry that picks up at the back of the ballpark. I suggest checking schedules before taking the ferry. Some ferries leave anywhere between 15-30 minutes after the last out.