We transported our dog from UK to Australia two years ago. Qatar's website gives details on shipping animals, so I thought it would be easy enough to arrange with them directly rather than using an animal transport company. Unfortunately, when I contacted them (and being put through to various teams), I was eventually told that I would need to use a transport company. I think that because of the complexities of administration they don't want to take the risk of dealing with the general public.
You can speak with them, but you may end up finding you need to use an animal transport company.
I'm pretty sharp on paperwork, legal requirements and such - but in the end, I was glad we used an animal transport company. It really is a logistical mess that would have taken me many hours and with every chance of screwing something up along the way.