Pixel & Tonic's Contact Form Plugin - POST submissions to Zapier Webhook URL?
By putting the webhook URL in the action field, you're completely bypassing the Contact Form's processing of the POST request and getting back a JSON response from Zapier.

I believe the correct way to do this would be to write your own custom Craft plugin that listens to Contact Form's beforeSend event.

When that event gets fired, you can make an AJAX request (using Guzzle or whatever) to Zapier's URL passing in the required fields, get the response back, do with it what you will, then let the Contact Form's controller finish executing and send them email.