Allocate space for object 'dbo.SORT temporary run storage

This is pretty self explanatory - your tempdb is full, and can't be grown to perform the operation it needs tempdb for.

As indicated, either the disk where it is located is full, or you haven't got suitable autogrowth settings (see this earlier post for information about that), if it's not autogrowth, you'll have to either delete other files from the disk that your tempdb is on, or locate your tempdb on a bigger disk.

This sqlservercentral post is similar to yours and might take be worth a look. This video by Brent Ozar might be worth a look too, covering tempdb performance and config.

Run this command to get the estimated tempdb space required for the check:

DBCC CHECKDB ('databasename') WITH ESTIMATEONLY

See the DBCC CHECKDB documentation

ESTIMATEONLY
Displays the estimated amount of tempdb space that is required to run DBCC CHECKDB with all the other specified options. The actual database check is not performed.