Getting a record count per user is possible. It would require two tables. One for the list of users, and one for the transactions. You would link the user ID to each transaction so that the transaction records would display in the related field in the users table. You would then add a Count field type on the transactions field in the user table.
For the record limit, there is not a feature in either platforms (Airtable or Softr) to accommodate this requirement. In my opinion it would take a combination of automations and API fancy-work
Submit an enhancement request to Airtable and Softr. You never know, it may happen some day.
I guess you could pass as hidden field who adds entries then du Airtable formula to calculate number of entries per user and then define user groups like one which is above limits hide the form and show some message