How to check for a local part on the server?

If the "local part" is created by the client, it will simply not exist on the server due to properties of Filtering Enabled.

The part will only exist on the client. If you want to check if a part exist on both the client and the server you can use a RemoteEvent to send the part object and check if it's nil or not.

--server
RemoteEvent.OnServerEvent:Connect(function(player, part)
    if (part) then
        print(part.Name .. " exists on server.")
    else
        print("Part doesn't exist on server.")
    end
end)

--client
RemoteEvent:FireServer(workspace.Baseplate)
RemoteEvent:FireServer(Instance.new("Part", workspace))

Output:

Baseplate exists on server.

Part doesn't exist on server.

Good luck.

for _, Model in pairs(workspace:GetChildren()) do
    if Model.Name = "ButtonModel" then

    code
     else
       code
end
end