This is definitely something that you can do with Zapier. You’ll need to use a service/app that sends SMS messages to customers, you can find all of the phone and SMS apps supported by Zapier on this page. It’s important to note that SMS by Zapier (our in house texting option) can only send text messages to one number so it wouldn’t work in this case.
Once you’ve found an SMS app you like, you can set up a Zap that automatically sends a text when the Gravity form is submitted. If you find yourself asking ‘Can I do X with Zapier?’ this article can help with that: Will it Zap? How to work out if Zapier can help you with your workflow.
I hope that helps, please let us know if you have any questions!