Your question involves two different issues:
1) How to launch script from a button?
2) Can your API be called client side from QuickBase?
To answer the first question see the attached pdf named "Button Study" which demonstrates six different ways to attach a script to a button. Since you are new to frontend you may want to just use the first method as the remaining methods deal with more advanced schemes.
To answer the second question your API needs to support CORS (Cross Origin Resource Sharing) or return JSONP (JSON with Padding). It would help if you could identify the API and point to the docs.