How are api requests identified client side?

I have a website, using a third party api service that is relatively open but still has obvious restrictions (i.e. if a person is sending 10,000 requests per second they will get blocked).

I have functions that are on the frontend that call the api so anyone can modify the code to call 10,000 requests per second. My question is, will the requests show up as mywebsite.com sending requests or as the ""hackers"" ip address. Will my website be blocked from using the api service if somebody does this?