I encountered the same situation several years ago after moving a back-end database from Access to MySQL (keeping the front-end in Access). I wound up using the following workaround:
Dim cdb As DAO.Database, qdf As DAO.QueryDef, rst As DAO.Recordset
Dim lngCustomerID As Long
Set cdb = CurrentDb
' create pass-through query to insert new row and then retrieve the IDENTITY value
Set qdf = cdb.CreateQueryDef("")
qdf.Connect = cdb.TableDefs("customers").Connect ' get .Connect from existing linked table
qdf.ReturnsRecords = False
qdf.SQL = "INSERT INTO customers (customerName) VALUES ('GordCo')"
qdf.Execute
qdf.ReturnsRecords = True
qdf.SQL = "SELECT LAST_INSERT_ID()"
Set rst = qdf.OpenRecordset(dbOpenSnapshot)
lngCustomerID = rst(0).Value
Debug.Print "LAST_INSERT_ID() returned " & lngCustomerID
rst.Close
Set rst = Nothing
Set qdf = Nothing