Are there any large flying creature transports? I wanted something for the party to fly around on that wasn't mechanical in nature.

If not in the books, in expanded star war products?

The Zillo Beast was carried by cables and gunships before beign taken to a Venator , for what it’s worth.

There are flying mounts on Onderon, I think they might carry two people at most.

Doesn't SWTOR feature thrantas on Alderaan? That'd be fun.

As SavageBob said, you've got thrantas on Alderaan; and there's no reason they couldn't have been transplanted off world before it got blown up.

In the video game Galactic Battlegrounds, most of the gungan 'vehicles' were actually living creatures; and their air transport was something called a "Dunsenn FlopCarrier"

It’s conceivable that you could use Loth Bats. (If you’ve seen Rebels)

Purgills are also a possibility. Although a potentially deadly one.