It is possible, but they don't recommend it, seeing as the Peninsula Papagayo Activity Center Clubhouse only has a few parking spots. In addition, driving there is not self-explanatory, I'd recommend using the shuttle once, then figuring out how to get there.
We were also going to drive ourselves there, but ended up taking the shuttle every day. The nice thing about the shuttle is that the golf carts are ready for you when you arrive at the Peninsula Papagayo Activity Center Clubhouse. If you arrive on your own, there's a chance that there won't be golf carts around, and you'll need to wait for the 'last mile' pickup to get to the beach house.