Can I play GTA Online if my disc is unreadable?

I own a physical copy of GTA V but it is too scratched to install on my son's Xbox 360. How do I prove ownership so he can just play GTA Online?

I'd contact Rockstar Games Support to prove ownership and have GTA V / Online activated on your Xbox 360.

Note, however, that starting December 16, 2021, you will no longer be able to play GTA Online on Xbox 360 and PlayStation 3. Even if you prove ownership to Rockstar Games Support, you will not be able to play GTA Online on or after that date. Only GTA V Story Mode can be played on or after that date. Currently, you only have about two weeks to play GTA Online on Xbox 360.

According to the Rockstar Games announcement post, GTA Online for PlayStation 3 and Xbox 360 Will Shut Down on December 16, 2021:

...the PlayStation 3 and Xbox 360 versions of GTA Online, including website stat tracking via the Rockstar Games Social Club, will officially be shutting down on December 16, 2021.

Please note, these changes are strictly for the PS3 and Xbox 360 versions of GTA Online, and will have no impact on access to, or progress within, the Grand Theft Auto V Story Mode.

From the Rockstar Games Support article, End of online services for PlayStation 3 and Xbox 360 versions of GTAV, Max Payne 3 and L.A. Noire FAQ:

GTA Online on PlayStation 3 and Xbox 360

• September 16, 2021
  • PlayStation 3 and Xbox 360 versions of Shark Cards for GTA Online no longer sold.
• December 16, 2021
  • GTA Online services for PlayStation 3 and Xbox 360 versions, including website stat tracking via the Rockstar Games Social Club no longer available.
  • GTA V Story Mode for PlayStation 3 and Xbox 360 not impacted.