First of all, I want to share some advice from someone I spoke to on the Player Behavior team regarding sending manual reports via Support tickets:
While we're not always able to take action, we definitely do when it is necessary. We don't want to set the expectation that players can use Support as a venue to report, because that's not what Support is. We are here to help explain account standings, help with account issues, explain how our systems are working with the account, and send players to the correct channels to empower them on fixing the issue.
That being said, if you have information about a person that falls outside of what the Instant Feedback System can deal with, you can submit a support ticket and talk to them about it.
https://support.riotgames.com/hc/en-us/requests/new?ticket_form_id=4584
An example would be the ticket I just sent in a few minutes ago regarding a player who has a racial slur in their summoner name. I didn't actualy play a match with them, so I cannot report them that way, but I just so happened to see it in a screenshot posted here in the forum and wanted to give Riot a heads up. Another example would be if someone is being toxic in a pre-game lobby and then someone dodges, and you have screenshots of their behavior. Also, if someone sends you a friend request after a match and then harasses you via private messages.
So in a nutshell, you would only send a ticket if you cannot use the report feature, or if what someone is doing is REALLY REALLY bad like death threats or a mad girlfriend posting your address, phone number, and other personal information in matches. If someone has gone to some extremes to harass you in a match and it will make you feel better to report them manually in a ticket, then go for it. Support will still be there for you, and you can also try to have a productive conversation with them about how to deal with this type of behavior.
Whatever you decide to do, best of luck to you in future matches!