The short answer is No. The side port or the 10 pin connection provide communication with the camera and allow it to record the GPS data. The hotshoe may be used to hold the GPS device, but it still needs to be connected. Even the third party models are plugged into the camera.
There are plenty of GPS devices that can be used and data imported after the fact. That's a little more complicated.
The RRS L-bracket has a slot and can be repositioned to provide more space. There is also a 3rd party USB L connector that could be used.