Which muscles are most important in holding the upper body "up" while it is leaned forward towards the ball at address? I'm guessing back muscles but it's not my area of knowledge.

a bit on your posture. If your posture is correct and you don't slouch, you're lower back will be busy but the muscles that actually 'hold the upper body up' (i.e. stop you from trying to touch your toes) are primarily the gluteus and hamstrings.

Peter

Yes, Dear This is right answer, Back muscle makes a very important role in our body to holding up...

your front side abdominals are stabilizing the core.
so if your posture, the front make sure you stays stable not just the back hamstrings etc..
abdominals function to hold things.