I have seen a lot of small towns finance these kind of things with a combination of fundraisers (potlucks, bake sales, door-to-door, etc...), straight donations, and one or two major benefactors. If you have a Main Street organization in your city, or in one nearby, they are good ones to chat with as their organization is really experienced in it. Community foundations also love this kind of thing, especially if there are other partners and they are just helping get across the finish line (not carrying the entire weight).
Fundraising is actually a good bar to cross to make sure you have community support. Perhaps the most important. North America is littered with community centers that were built in the hope of creating community where none existed, only to discover that a community center only works when there is a community to fill it. I'd make sure I knew what shortcoming the supporters were trying to address and see if there is a minimum viable product you can field test to make sure you're on the right track.