Is there any way to do this? Some people on a server I host would like to have the option to not broadcast their location via the player icon in Rei's Minimap.
No, there is not a way. You can tell everyone to turn off their Entity Radar for players, but as Rei's is client side only, you cannot force anything off the map.
Well, as the server operator, you can disable player display on the minimap altogether with MotD settings (you'll need to add a mod that does proper motd output; rei provides one)
At that point, for them to see player position they'd need to actually edit the code for rei's minimap, and most players either don't know how, or wouldn't go through that effort on a private server when it's clear the admin has chosen not to.
If your not the server admin you can always make the other players hate the feture so they request its removal.
Stalk players like they were any other type of mob. I suggest doing this with a vorpal weapon so you can put there heads on pikes (fence posts) around spawn.