According to this community response, it is possible, but you cannot easily ingest and index the data from the on-premises SQL Server from the Azure Portal UI (Import Data Wizard). Instead, you'd have to create a process to use their API to get that on-premises data into the Azure Search indexes.
If you want to do this today in SQL 2022, you would have to roll quite a bit on your own. For the calls to Open AI, you can implement REST calls with help of CLR procedures. To make use of the data that comes back, you would have to implement your own vector-handling.
If you can be little patient, there are several new features in this space in the upcoming SQL 2025. There is a built-in stored procedure for the REST calls, and there is also vector support, so that you can match the response to the user query with what you have previously have received for whatever you had in your database.