Xbox Live Access Issue

Sorry to hear it. If your son has a shared subscription from a different account then that account would either need to be logged in or set as Home on his new console. https://support.xbox.com/en-US/games/game-setup/my-home-xbox

My son got a new xbox and he signed in to his account, but it is not recognizing his xbox live subscription. It keeps telling him he has to purchase a subscription but we already have an active one that is good until December.

Might redownload his account

Verify that the account logged in to the console is the same account that has the subscription service.
On the old console has the account been logged out?

If his account bought Live it must be shown on the new console. Subscription is connected to account, not to console.

Try a hard reset by holding down the power button for 10 seconds until the console shuts off.

Unplug the power cable and wait for 5 min before plugging in back in.

That might fix the problem