Is it possible

Is it possible to create a campaign for CPC to drive traffic to a button after people fill out a form on my website and only pay when someone clicks that button?

No, it's not possible, not with cpc. However, this is doable with cpa/cpl type of advertising.

Google adwords allows you to do a lot of things like adding a phone number to the ad and providing a button to call the phone number directly from the ad also you can make use of sitelinks feature to direct traffic to the specific page with your form.
Cheers!