There are only two ways to restrict who fills out the form: by requiring a password (same password for everyone) or by restricting users to a particular domain.
You can prefill forms, but not with different values based on who is logged in.
You can configure the form to collect whoever filled out the form in a Created By field. The toggle for this feature is in the settings at the bottom of the form configuration. The user must be logged into Airtable in order to get this info.