In order to use Zapier with Squarespace, you will need to be on the Website Business plan or higher. To learn more, visit their Premium features page.
Presently, it is only possible for Zapier to receive data from Form Blocks in Squarespace. The forms you can create inside of a Cover Page are a separate entity and are not yet supported.
As a workaround, you can configure your Cover Page form in Squarespace to use Google Drive or an email address as its storage. You can then trigger your Zap using the Google Sheets or Email Parser integrations respectively!
Squarespace only allows for one live Zap to be associated with a form, so if you have an existing Zap turned on that's connected to your form, you'll need to turn it off before turning on another.