Can I process an invitation for a private visa through a travel agency?

Unfortunately you cant. Travel agencies can provide you only with a tourist or business invitation.

No–travel agencies can only work with tourist or business invitations for visas. Private visas are the most difficult to get and can only be obtained via invitation from family or friends who live in Russia or the Ukraine.

If you travel to Vietnam and need a private visa then you can use Vietnam visa on arrival and a travel agency can help you to get it. Check the travel agency for more details https://www.hotels-in-vietnam.com/