Considering how all the usual docking/berthing ports at the station are occupied, Santa would likely arrive in a Dragon Crew/V2 or a Boeing CST-100 vehicle. These are the next US vehicles that will be manned and due for first launches in 2019.
The Russian side uses their own style of docking port that Progress, Soyuz, and the ESA ATV vehicles dock. These can dock automated or manual and there are 4 of them on the Russian end of the station.
The US side of the station has 2 Berthing ports, currently occupied by the Dragon (Cargo) and Cygnus (Cargo) vehicles. These cannot automatically dock and require on station help. They use the CanadArm2 to capture the vehicle and pull it in to the port. The opening and closing of the internal door is manual from the inside of the station. (So you could not leave an unmanned station this way, nor arrive.)
The two new vehicles will use the docking ports on PMA-2 (at the end of Node 3) or PMA-3 (facing upwards on Node 2). (PMA-2 is how the Shuttle would dock, and they added an IDA (International Docking Adapter) to it to allow Dragon/CST-100 to dock).
These ports allow automated docking and undocking.
So 2 docking ports are still open. Cargo ships are unmanned, so he would have to use a port that allows manned vehicles.
A nice question showing the available ports is Will The ISS Need more ports?
From that question, here is a nice picture showing one configuration from the past where most of the ports were occupied, like the picture in the question.
That picture is from 2011. As of end of 2018, the up facing port the HTV is connected to will be PMA-3 for manned docking of CST-100/Dragon vehicles, and the Shuttle is docked to the PMA-2 port.
The lower port, just below the HTV is where the Cargo Dragon is currently berthed, and to the left of it, there is the old location for the PMM module that has since been moved to Node 3 (with the BEAM opposite it) make space for where the Cygnus is currently berthed.