This is almost certainly it. When a user running a Laserfiche application installer attempts to register that application through LFDS, the wizard makes a Download Application License call to LFDS under their identity. Thus, the user must have the Download Application Licenses LFDS permission or they'll receive an "Access Denied" error like you see here.

I am in there now and I see the user has already been granted rights to do everything including download application licenses. This would make sense as they have installed quickfields before.

The only thing that I can see that looks off to me is the user logs into Windows using a different username than the one displayed in Directory Server. I think Active Directory supports multiple usernames assigned to one user SID now to allow for last name changes as people get married. Could this be why it thinks her access is not allowed? I don't think I can choose which username Directory Server uses. I am not even sure if this is the problem, but not sure what else it would be.

Perhaps you could test that theory by trying to add the username they log into Windows with as a new Windows User in LFDS. If you get a "User already exists" error message, that's a good sign LFDS is associating their multiple usernames with the same SID. If it goes through, that would indicate LFDS is seeing them as separate user accounts, and I would look at both user profiles in LFDS to see if they have different UPNs. 

Yes, we get     
Could not register new user: Object already exists.

When I am trying to add the user, I see the domain name they are using to login to, but in the user list I see their old name they no longer use.

Hi Chad,

Does the user have "View" and "Add objects" rights to the organization they are registering the application under, as mentioned in this post?

Every single box is checked under Rights Assignment. They have it all

Temporarily put the user's AD account in the local "Directory Server Administrators" Windows group on the LFDS server, which grants all possible rights in every area. If that still doesn't work, you have some weird AD mismatch going on that's not worth the effort of solving for the sake of issuing a Quick Fields license.

Manually generate the QF license in LFDS with "+Applications" and provide the lf.licx file to the user so they can provide it to the install wizard instead of selecting the LFDS registration option.

Don't forget to remove the user from the local Directory Server Administrators group.

Either way, you might want to try deleting the user's existing AD account registration out of LFDS and re-adding them under their new name as a precautionary measure against future weirdness.