Is there a meaningful concept of a mountain having a footprint area?
The mountain you see is only a part of the story. From a geological view all mountains have roots, in which the weight of the mountain depresses the upper crust into the lower crust (occasionally with gravity inversion, as in salt diapirs), and the overall crust is depressed into the upper mantle. This results in a regional gravity anomaly which could be regarded as one form of the mountain footprint. That is, a map of the deviation of the mountain's 'Bouguer anomaly', away from the regional gravitational envelope.
But that's just a geophysical perspective. You could doubtless develop comparable 'footprint definitions' from a hydrologic or geomorphologic perspective.