Overwatch won't let me send friend requests to anyone

Check two places. In game, options then social, there's a setting for who can send you friend requests and a couple of other communication toggles, make sure yours aren't set to nobody. Then check the Blizzard account itself, if it's flagged as a child account or has parental controls on, friend requests get blocked at the account level and the game just fails quietly. If you're on console it can also be the console's own privacy settings for communicating with others. One of those three is almost always it.