How to route every API request through own proxy?

On my LAMP server, I have a Javascript, JQuery based app, which makes requests to a remote API and displays its responses. As long as only I use the app, everything is OK. But now I must open this app to multiple users, and if the API gets multiple requests from the same IP, it bans the IP.

What is the way to route each request through its own proxy address? I have many working proxies as a list, and a proxy gateway, which can route multiple requests.

A simple example or just a correct keyword to search would be enough as an answer.