Zapier makes it a lot easier since you don't have to write an API integration script yourself.
Through the Zapier panel you can create a zap that receives data from a webhook and passes it on to Salesforce.
The only thing left is to get a JS script that would take the form data and push it to the Zapier webhook.
Unfortunately, this needs to be custom coded.
Any developer should be able to knock this out in a few hours. Maybe less but depends on how many tests need to be run to make sure everything is working as expected.
Depending on your monthly volume of form submits, you might want to consider using a custom script for the integration and skipping Zapier altogether.