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The dealer has to plug a Tech 2 into the vehicle to program the remotes.

It appears vehicles using the old style BCMs can have remotes programmed without a scan tool, and GM publishes the non-scan tool procedures, as well as procedures on how to program with a scan tool, in the service manual. For the H3, there is only one procedure and that is with a scan tool.
Any GM dealer should have a Tech 2, and it should have Hummer information on the tool. So any GM dealer should be able to program the remote, or an independent garage with a universal scan tool that can access the H3 BCM should also be able to program the remote.
Programming time is about three minutes, if the tech is familiar with the Tech 2 menus.
If you know someone with a scan tool, here is the procedure:
Transmitter Programming

Important
Up to 4 transmitters can be programmed. Do not operate or program the transmitters in the vicinity of other vehicles that are in the keyless entry program mode. This prevents the programming of the transmitters to the incorrect vehicle. The order in which the transmitters are programmed is important. The first transmitter programmed will be transmitter #1, and the second transmitter programmed will be transmitter #2. Use care to program the transmitters correctly.



Important
All key fobs must be programmed each time any one fob is programmed. The scan tool must stay connected until all of the transmitters are programmed.


Install a scan tool.
Turn ON the ignition with the engine OFF.
Build vehicle in scan tool.
Select Body.
Select type of radio.
Select Body Control Module.
Select Special Functions.
Select Program Key Fobs.
Follow the scan tool on-screen directions.
After programming the last transmitter, remove the scan tool.
Exit the vehicle. Attempt to lock/unlock the doors to verify the programming.
Operate the transmitter functions in order to verify correct system operation.