Using experiments and simulations, we study the flow of soft particles through quasi-two-dimensional hoppers. The first experiment uses oil-in-water emulsion droplets in a thin sample chamber. Due to surfactants coating the droplets, they easily slide past each other, approximating soft frictionless disks. For these droplets, clogging at the hopper exit requires a narrow hopper opening only slightly larger than the droplet diameter. The second experiment uses soft hydrogel particles in a thin sample chamber, where we vary gravity by changing the tilt angle of the chamber. For reduced gravity, clogging becomes easier, and can occur for larger hopper openings. Our simulations mimic the emulsion experiments and demonstrate that softness is a key factor controlling clogging: with stiffer particles or a weaker gravitational force, clogging is easier. The fractional amount a single particle is deformed under its own weight is a useful parameter measuring particle softness. Data from the simulation and hydrogel experiments collapse when compared using this parameter. Our results suggest that prior studies using hard particles were in a limit where the role of softness is negligible which causes clogging to occur with significantly larger openings.