Answer:

You can only use ODBC Administrator button on the DSN tab of the Foxhound Menu if Foxhound is running on the same computer as the web browser; otherwise, the button is disabled.

This restriction is a practical one: The ODBC Administrator button starts the ODBC Administrator on the same computer as the web browser, and it only has access to DSNs located on that local computer. If Foxhound is running on some other computer on the network, it can only use DSNs located on that other computer, which makes the ODBC Administrator button on the local computer useless... and that's why it is disabled.

Workaround: Use the String tab of the Foxhound Menu to create and edit connection strings, instead of using the DSN tab. The connection strings on the String tab are available to you for editing via a web browser running on a different computer from Foxhound, and they are stored inside the Foxhound database so they are available to Foxhound for making connections to target databases.