Where do pending drush database updates live? I'm using drush to update the modules on our drupal site. I updated the video_embed_field module to the latest version.

This update required 3 database updates to occur, 7006, 7007, and 7008. For some reason 7006 failed as documented here. That link includes a patch which should fix the problem, the only issue is I have no idea where the patch needs to be applied. i've applied patches to drupal modules before, but video_embed_field does not appear to have a directory in sites/all/modules which is where I think the patch ought to go.

When I run drush updb it wants to apply those three database updates, but I can't find them stored anywhere in the file system (or any references to video_embed_field at all). Are these database updates stored in a table in the database, is their a folder somewhere I'm not looking at? Is there another folder where modules get placed?
Drupal will look in /modules and /sites/*/modules if the site was installed on top of the default profile. If you're using a distribution or other contrib/custom profile then it'll also look under /profiles/[name-of-profile].

SELECT filename FROM system WHERE name = 'video_embed_field'


will tell you exactly where Drupal is looking for the module you're trying to find.


  Are these database updates stored in a table in the database[?]


I haven't looked at Drush, but as far as core's update.php goes, no. The .install file of each enabled module is checked to see if it has any update functions with a schema greater than the currently installed schema. I'd imagine Drush does the same thing, no real reason for it to do things differently.

If that assumption holds, then you basically couldn't be seeing those updates as options unless the module is somewhere in the file system (or at least a PHP file that will be included in a normal bootstrap containing a function called video_embed_field_update_7123() or similar).

So keep looking, it's likely to be there somewhere!