They don't grant requests for chat restrictions, and even if they did, if you abused your limited chat anyway you'd just get pushed up to an account ban because you asked them to put you there on the punishment tiers. Chat restrictions still allow you a certain number of messages throughout the match so that you can have some verbal communication with your team.
Instead, I recommend utilizing muting far more often. I see most often that someone is in a toxic argument with someone when they get punished, and that if they had only muted earlier they'd still have their account.
If you are the one instigating the argument because of their gameplay, there isn't much you can do to make them play better. Try to realize that they are more likely to play worse than to all of a sudden become a better player because you told them to. If they are trolling, they are probably getting a kick out of your response, so either way it's better to cut your losses and not respond to them.
You can still forcefully cut your chat off, though I wouldn't really advise it over just muting people. You can rebind your enter key with a customizable keyboard or other software like autohotkey (so that you just can't open the chat bar), or you could move your chat window off the screen entirely, or even shrink the chat window's size down to nothing manually in the config files.