"Y" in this context is short for the acronym "YMCA" (which itself is short for Young Mens' Christian Association, but few people ever call it that). The YMCA provides athletic facilities like gyms and swimming pools to the public at low cost.

I'm not American, nor am I familiar with the Frog Tree House, but I know that Y is sometimes short for YMCA. Here in the UK they work with homelessness, addicts and youth work, but I believe in the US they also have sports facilities, including swimming pools and gyms. Maybe here too, I don't know that much about them. It originally stood for Young Men's Christian Association, but I here it is not exclusive to Christians. You may need an American to confirm, but I hope this helps.