How would one add a model or hat to a character when they spawn?

First of all, make sure to put an effort towards trying some stuff out first and telling us how it goes.

Besides that, what you're asking is relatively simple.

Simply make a script in ServerScriptService and put the following code into it:

local model = PutTheModelLocationHere
local team = PutTheTeamLocationHere

local players = game:GetService("Players")

players.PlayerAdded:connect(function(plr)
    plr.CharacterAdded:connect(function(char)
        if plr.TeamColor == team.TeamColor then
            model:Clone().Parent = char
        end
    end)
end)

If all goes well, this should work out for you!

If you have any troubles with the script provided, please comment the error messages from the output and I'll try to resolve it for you.

EDIT: Make sure you put the model and team location in correctly