About asking for reports....

here is your answer https://support.riotgames.com/hc/en-us/articles/201752884-Reporting-a-Player

REPORTING GUIDELINES Pro tips: Make sure your report is tailored for the behavior you saw. It may be tempting to select every report category available but this is unnecessary. It only takes one report to trigger a review by our system selecting more than one category will do nothing to make your report more valuable.

When possible, leave a concise and clear comment about the behavior you observed. For example, if the player decided to throw the game by rushing down middle lane after their first death a good comment would be: “Died at 5 minutes to a gank and then proceeded to run down mid for the rest of the match”.

Things to avoid Do not threaten or repeatedly tell a player you will report them. Doing so can encourage players who are already negative to continue their behavior. Whether they know they are being reported or not has no bearing to whether the system will act on them. But most importantly, repeatedly threatening or arguing with a negative player can end up derailing the game for everyone else and then open yourself up to reports and possible disciplinary action as well. Avoid negative thoughts and useless chatting with poor performing teammates! Focus on victory by muting the offending player and then reporting them at the end of the game.

Do not ask other players in the match to report the offending player. It only takes one report for our systems to review a game. Additional reports will not do anything for the offending player; however as mentioned above, it could open yourself up to a report of your own; especially if you are derailing the match by constantly demanding reports of other players.

Negative behavior is unacceptable regardless of the circumstances or who started it. It can be tempting to respond to negative players with negativity of your own. Our systems do not care who started the behavior and will treat each reported player based off the merits of their own case. The only way to handle these types of players is to mute them and continue playing the game followed by giving them a report at the end of the match

I believe it's seen as toxic by the system so would help contribute, but I think it's unlikely you would get a ban or restriction from that alone.

bannable or not it's completely unnecessary

it takes exactly 1 report to trigger a review

{quoted}

Does anyone know if asking for reports is bannable?

It is not if you do not ask for it that much, just dont spam it.