One method would be to use GlideAjax in a Catalog Client Script and send the data from the form to a Script Include with your REST API script. You would have to run the REST API script in the Script Include because the Catalog Client Script will not run on the Server.

Personally, I would create a Run Script activity that has the REST API script and put it in the workflow of your Catalog Item. That way it will easily run once the Catalog Item is submitted and you can easily access all the variables from the form. A workflow activity runs on the Server so you can put the REST API script or just call a Script Include.