I'm not receiving any test data
You can create test leads with a click of a button through the Campaign Manager UI. The complete instructions are mentioned in this help article.
My Zap stopped receiving leads, but it's still turned on
When you connect an account to Zapier, we receive an authorisation token that's used every time we try to connect to that app. The authorisation tokens for LinkedIn expire after 60 days. This means that you will need to reconnect your LinkedIn Account at least every 60 days.
To test whether your account is still connected, go to your Connected Accounts page, find the LinkedIn connection and click 'Test'.
To reconnect your Account, click on the 'Reconnect' button (you can do this from your Connected Account page, or in the Zap).
What permissions do I need to access data via the New Form Response trigger?
LinkedIn offers a guide here detailing what permissions are needed for access of this data.
In short, you must be assigned as BOTH the Account Manager or Campaign Manager role on the ads account in the Campaign Manager tool, as well as the Company Page Admin or Lead Gen Forms Manager role.