How can I convert this from a LocalScript to a regular script that works in ServerScriptService?

Put a RemoteEvent in ReplicatedStorage named "UpdatePet", if you change the name of it change the name in the lines that start with "local RemoteEventName = "

In the local script you had before:

local RemoteEventName = "UpdatePet"
wait(2)
game:GetService("ReplicatedStorage"):WaitForChild(RemoteEventName):FireServer()

In a server script in ServerScriptService:

local RemoteEventName = "UpdatePet"
game:GetService("ReplicatedStorage"):WaitForChild(RemoteEventName).OnServerEvent:Connect(function(plr)
	local petname = plr.Pet.PetName.Value
	local mesh = game.ReplicatedStorage:FindFirstChild(petname)
	-- local fly = plr.Pet.DoesItFly.Value
	-------------------------------------------------------------
	
	local part = Instance.new('Part', workspace[plr.Name].Torso)
	part.CanCollide = false
	part.Anchored = true
	part.Size = Vector3.new(1,1,1)
	mesh:Clone().Parent = part
	local direction = CFrame.new(part.Position, plr.Character['Left Leg'].Position).lookVector
	local increment = direction * 4
	while true do
	    wait()
	    part.CFrame = game.Workspace[plr.Name]['Torso'].CFrame + (direction * increment) + Vector3.new(-5,0,0)
	    part.Rotation = game.Workspace[plr.Name]['Torso'].Rotation + (direction * increment) + Vector3.new(-1,-2,increment)
	end
end)

I didn't test this, if it doesn't work let me know and I'll look into it