How can I make client start without having turned off firewall?

I think that you might be able to go into your firewall settings and whitelist the game client. Not entirely sure how to do it though, because it's been months since I last looked up how to do something like that.

Try searching around the internet for how to whitelist league in your firewall, that may help.

You just need to add it to the rules as an exception, but the firewall should have asked you about this the first time you ran the client, it ask you about both private and public networks, public networks is disabled by default, if you didn't mark it and you are currently on a public network, or simply your network is configured as public, then it will block it in those cases, adding the rule is easy if that is not the case.