We experimentally study the flow of oil-in-water emulsion droplets through a quasi-two-dimensional hopper. The hopper chamber is sufficiently thin so that the droplets are deformed into pancake shapes. Due to surfactants coating the droplets, they easily slide past each other, approximating soft frictionless disks. We find that clogging at the hopper exit requires a narrow hopper opening only slightly larger than the droplet diameter. This is in contrast to previous studies with hard disks that found arch formation and clogging with significantly larger hopper openings. We conduct simulations demonstrating that softness is the key factor reducing clogging for emulsion droplets. With stiffer droplets or a stronger gravitational force, clogging is easier.