We have a free event that requires participant approval. We are manually approving and sending confirmations to those who are allowed to attend, however there's some confusion because a message pops-up during the process that says they will receive an email to complete their registration, and that doesn't happen (we don't need it since there's no payment and we've already collected all information). We already have this option turned off, but the message is confusing registrants. Where do I find this message to edit/override it?
Sounds like a job for grep. Use
grep -r "Once approved, you will receive" templates in your civicrm folder, and it will tell you it's templates/CRM/Event/Form/Registration/Confirm.tpl. Similarly grep'ing for the other one it's in ThankYou.tpl in the same folder.
On the 'Online Registration' tab of the Event configuration is an option to select "Send Confirmation Email"
Sounds like you want to set that to No.
However I think that will still result in an onscreen messaging saying "Your registration has been processed successfully." which sounds like it would still be incorrect.
If this is the only event running you might be able to use Word Replacement to try overwriting that. Otherwise it would require a tweak to the template I think