Setting up forms with a Web API Call

I am currently trying to set up a Form on my website that collects a zip code (string value) and a mile radius (byte value) from the user, and then retrieves information based on those values via a Web API call from a 3rd party website and loads that information back onto my page.

I have no clue how to do this. Does anyone have any helpful information or articles they can share? I was really hoping there was a plugin out there to simplify this, but I haven’t found anything yet.

*Additional info … this is for a “store locator” type of search on my website. I have permission from the 3rd party website to send the Web API Call, and they provided me with a unique token. I would ask them for information on how to set this up, but they seem to be a bit busy, and I’m hoping to figure this out somewhat quickly.