>>What error dou you get back?
>
>>SQLEXEC() returns a -1.
>
>Further, AERROR tells me the error is 1526, which indicates an ODBC error. Again, this same code works without problem if users aren't logged in. Yet an ADO recordset can be created via ODBC regardless of whether users are in the application or not.

First, I think, you have created the server-side ADO recordset. If you'll try to create a client-side recordset or retrieve records from server-side recordset, you'll probably get the same error.
Second, IMO you have installed Jet 4.0 version of Paradox ISAM driver. This version provides read-only, exclusive access to Paradox files. To get a full functionality you need to have:
Jet 4.0 ISAM driver + BDE installed
OR
Jet 3.51 ISAM driver
OR
Jet 4.0 (Service Pack 5) ISAM driver.
See details in KB articles Q263561 and Q230126.