How would someone go about making a "Tag" function?

You should probably just check the distance between players instead of having each of their parts have a touched event and if you want it to be similar to jailbreak.

local runService = game:GetService("RunService")
local contextService = game:GetService("ContextActionService")

local ply = game.Players.LocalPlayer
local char = ply.Character
local ui = ply.PlayerGui.TagUi	--Assuming this is a ScreenGui

ply.CharacterAdded:connect(function(c)
	char = c
end)

function tweenUi()
	--Put Code to tween your Ui and tag progress 
end

function searchTarget()	--Check if player is in range
	local dis = 3
	local tar = nil
	
	for _,v in pairs(game.Players:GetChildren()) do
		if v.Character and v.Character ~= char then
			local tDis = (v.Character.PrimaryPart.Position-char.PrimaryPart.Position).magnitude
			if tDis < dis then
				dis = tDis
				tar = v
			end
		end
	end
	if target then
		ui.Enabled = true
		target = tar
		contextService:BindAction("tagTween",tweenUi,false,Enum.KeyCode.E)	--Binding tag function to key
	else
		ui.Enabled = false
		contextService:UnbindAction("tagTween")
	end
end

runService:BindToRenderStep("findTarget",0,searchTarget)	--More functionality than while loop