Is there anyway to run seat.Changed or something similar on a local script?

I need to run a script when a seat that is copied locally has a player seated.

There certainly is. The way to do this, is everytime the seat changes, check the if the seat occupant is equal to the character Humanoid.

local plr = game.Players.LocalPlayer -- get the player
local chr = plr.Character or plr.CharacterAdded:Wait() -- get the character, if its not there then wait.
local humanoid = chr:WaitForChild("Humanoid") -- wait for the character humanoid

local seat = game.Workspace:WaitForChild("SeatName") -- wait for the seat

seat.Changed:Connect(function() -- when seat changes, perform this function
	if seat.Occupant == humanoid then -- if the humanoid is in the seat, run this code
		-- do whatever when they sit in it.
	end
end)

I hope this helps!