How to capture / digitise a live satellite stream
It looks like most of the content on Azerspace-1 is "in the clear", or not encrypted, which makes things easier. But there is still a lot of "it depends" for exactly what equipment you need. Aside from all of the earth station equipment (dish, LNA, down converter, etc.), you will need a DVB-S2 or MPEG-4 receiver. That could be a vanilla receiver with an external encapsulator or gateway.

There are definitely companies that provide teleport and satellite media services. I'm not familiar with Azerspace-1's footprint to give concrete recommendations. However, two examples of companies that might be appropriate are Intelsat (Fuchsstadt teleport) and Encompass.