As in a title, I'm searching for a way of reserving a place on a rocket to send small satellites into space. Can it be only done by some kind of launch broker company? Is there any information on how those companies work with SpaceX or any other companies providing satellite launch?
Generally, if you are only intending to fly e.g. one cubesat, the major launch providers won't be interested in dealing with you directly. It's not worth their time, and besides generally their primary customer "owns" a particular launch, so the space-available isn't theirs to sell.
As called2voyage mentioned, there is the NASA Cubesat launch initiative for nonprofits and universities that don't mind waiting a while.
Some up-and-coming small launch providers such as Rocket Lab may be an exception and willing to work with you directly.
If you want to launch by government owned space agency ISRO, then you need to contact its commercial arm Antrix corporation. You can obtain a launch slot in one of the PSLV rockets. For a CubeSat it will be a piggyback ride alongwith the main spacecraft. ISRO does not provide dedicated launches for small satellites. However, there are some private launch companies like Bellatrix Aerospace, Agnikul Cosmos and Skyroot, who can provide launch service only for CubeSats.