Hello Lil,
I'd like to help you on such a cause, therefore I can tell you that I could use the server I've got my websites to host yours without any charge for it.
As for the requirements (in here "we" means you and me):
1.- We can't "copy" the WIX code, but we can set the same layout and almost every functionality.
2.- The domain is available and the project bid includes it.
3.- I can set it up, and teach you how to improve it, keep it going.
4.- Quick loading of the whole site (images are quite important) is one of the priorities.
5.- We can add different forms as needed, if you want to I can show you how to make them.
6.- We could look into paypal integration or something similar, it would depend on your infrastructure.
7.- The site will be manageable with any need to code. So you can update, change, add, remove everything without needing to call programming cavalry.
On a personal note, here in Mexico we also support this kind of organizations, we even have two lovely dogs that come from rescue homes.
Feel free to tell me if there's any other issue you need to talk about before we can get going on this project.