Hello Bryan,
First, check that you are using the new "External Contacts" feature instead of the old one that it replaces. If you put your contacts into the old one, then blind transfer will not work. Here is an article on help.mypurecloud.com explaining how to access each of them:
https://help.mypurecloud.com/articles/two-external-contacts-features/
Once you know you are using the correct one, check that your agent has the necessary "externalcontacts:contact:view" permission (it is typically granted by the "PureCloud User" default role). If they do, then you should see section of results under the heading "External Contacts" when the agent performs a search while doing a transfer. You can see what that looks like in the UI here:
https://help.mypurecloud.com/articles/transfer-voice-interaction/
I hope this helps! Please let me know whether you are able to get it working.