After a bunch of searching, it seems to me that there's no way to move without potentially accidentally attacking champions. It seems asinine to me that such basic functionality doesn't exist especially in a game where positioning is like the single most important aspect. What are the chances of Riot adding this? Is a place to petition for such a thing?

Just for thoroughness:

• "Target Champions Only" toggle
  • Obviously doesn't work if you're trying to walk around a champion
• Disable "auto-attack" in game options
  • This only makes it so your champ won't automatically try to basic attack enemies you hit with an ability
• Hold down right click
  • You'll still attack anyone your mouse hovers over before releasing the click
• Aim "wide"
  • Not always an option, there can be walls, etc. that would make your character walk in a completely different direction than you want