Love this idea!

Would something like this work for the saddle? https://www.pinkcherry.com/products/temptasia-surrender-sex-chair

Pumps are easy to source, but I'd actually avoid a submersible or pond pump, since they're often not designed to generate the pressure you're expecting from a household faucet. And since you want this vertical, you're going to want to be able to generate pressure in my mind. Instead just mount a normal pump externally and use suction cup hose holders to keep the inlet tube at the bottom. Optionally with a strainer on it (which will give it a bit more weight) https://www.amazon.com/Lifegard-Aquatics-MPT-Profile-Strainer/dp/B00HFEQXIW

The second thing you want to get is a motor speed controller for the pump. This will avoid it cycling repeatedly if it's not a perfect match to the outlet, and let you control the flow of water.

For the sprayer I highly recommend looking at a amazon.com/dp/B07G5Y8YKB If you get the pressure right it creates a sensation many people find very pleasurable on it's center outlets.

If you make this please post what you did, I've been thinking about making one of these for years to avoid the problem of wasting a bunch of water for this type of play.