Antarctic firn is critical for ice-shelf stability because it stores meltwater that would otherwise pond on the surface. Ponded meltwater increases the risk of hydrofracture and subsequent potential ice-shelf collapse. Here, we use output from a firn model to build a computationally simpler emulator that uses a random forest to predict ice-shelf effective firn air content, which considers impermeable ice layers that make deeper parts of the firn inaccessible to meltwater, based on climate conditions. We find that summer air temperature and precipitation are the most important climatic features for predicting firn air content. Based on the climatology from an ensemble of Earth System Models, we find that the Larsen C Ice Shelf is most at risk of firn air depletion during the 21st century, while the larger Ross and Ronne-Filchner ice shelves are unlikely to experience substantial firn air content change. This work demonstrates the utility of emulation for computationally efficient estimations of complicated ice sheet processes.