I have a Webparts system to order on-line, but if I did not have it, I would go to the local dealer and give him part number.
You need to know your VIN, at least the first 6 digits. You can call them with part number and check if it is available before you go. If they agree to order via phone call - they will order and you just go and pick up. Or ask them to send it by courier. But if you have a part number and you do not know if it is available - call to check availability.
If you need to find a part, or order when you know it is available, you probably need to go there. They would tell you: "no, it is old car, we do not have it". Then you tell them:" move your a... and find it." Then they will find it and will tell you "forget it" or "it will take a while as we need to order through classic center" or we will have it within 24hrs" or "we will have it in 72hrs". Then you go and pick it up. At least this is how it works here.
Fluids, coolant - they may have on stock.
You talk to them personally to develop relationship and get a discount.
In the US you also have Tom in Classic Center, Irvine, who is a great help.