Read-only access only refers to people entering data through Airtable’s data layer or interface layer. It has nothing to do with forms or 3rd-party access.
All 3rd-party access to Airtable, such as Make’s ZenDesk integrations, communicate with Airtable through a designated user account.
So, you will need to make sure that your integrations are attached to a user account that has the ability to write to your database.
The only exception to this is Enterprise Service Accounts, which aren’t tied to any user accounts.