One of the most frequent hurdles video game club have for their creation is convincing the school district's IT department to drop some of the firewalls for the games your club will be playing.
In this section I'll talk about the various approaches different teachers I've talked to have convinced their IT departments to allow them access to the internet. Every district is different; the power structures and decision makers will vary, so you will have to figure out the appropriate way to go about this in your own scenario.
Start by going through The IT Department
I would recommend starting in this situation. Just give the system admin a call and talk about what you'd like to have available. Most districts block as many gaming domains as they possibly can, to ensure that kids aren't playing games during school hours. When I worked in Des Moines, this was no different. However, I was able to get the IT department to take down the firewalls for select games that I knew my students would be playing, but only during the time our club had our meetings. So, I had to give the system admin advanced notice of when we would meet, and which domains I would like him to open up. He would do it a bit early, so I could test that it's working, and when he returned from work the next day he'd close those domains again. This solution made everyone happy. I'd recommend starting there.
In one district I worked at, I had to go through each game we wanted to have available, and prove that they are not opening up themselves to viruses. The IT department was concerned that the games would download viruses that might compromise the entire school district's network. We walked through together all the kinds of packets and files that would be downloaded from the differing games, and the IT department eventually felt comfortable opening it up for us. This part may take some technical expertise, so if you aren't very familiar with how these work, hopefully it doesn't get to this point for you!
Go through the SUperintendent
It is not always the case that the superintendent can tell IT what to do, it all depends on the authority structure of the differing districts. I've worked in both kinds, one where the superintendent was able to tell IT to open their firewalls for our club, and one where IT fought back and said it isn't up to her what they do. So, if you're getting resistance from IT, perhaps try this route first. You may need to do convincing again about exactly why you think the video game club will be educationally valuable to the students, but you've already done a lot of that work by this point so it shouldn't be too much for you.
through the school board
If IT and the superintendent are being obstructionist about forming your video game club, and you're really in the mood to rustle some feathers, a last resort would be to go through the school board at one of their meetings. The superintendent ultimately serves what the school board asks of them, so you might be able to get the board excited about the possibility of engaging students this way, and they can apply the pressure to the proper people to get things moving.
Rally parental support to apply pressure to the others
A super final last resort would be to go to the parents and get them energized about it. Have them argue with you to the school board.
Be creative with your solutions! I've heard stories of teachers going through their PTO and rallying them. I had a teacher from Mississippi, who was receiving distinct hesitance from his superintendent about starting a League of Legends club due to the violence in the game, come to the superintendent and say, "Most cartoons and the Bible are much more violent than what I was seeing". His point was that even something considered sacred has quite a bit of intense violence, and the important part is to contextualize it through a responsible adult.