Off the top of my head, it seems like you could possibly do it if the other server allowed it.
One possible way would be to have the action go directly to their website (if they allow it).
<form method="POST" action="https://api.otherwebsite/upload">
Another possible way would be to have your own server that gets the request from the page, and then your server sends a post request with that data to an external api. However, if you are just trying to get data from another website, it would probably be easier to use your own server to get the data off the form, and then send a get request to the other server.
I would look at the Yelp api and see what they recommend for submitting a review.