Yes, you can use SSMS without having SQL Server installed locally. You can connect to any SQL Server instance you can find on the network.
You say that you want to connect to your production server from your development machine. I think you need to discuss how to do this with the DBA or your IT department. We don't know any details about your environment, so we cannot really help you with those details. However, it is not impossible that the network is configured in such a way that you cannot connect directly to the SQL Server instance from your development machine for security reasons.