One class of competitive candidates for dark matter is ultralight bosons. If they exist, these bosons may form long-lived bosonic clouds surrounding rotating black holes via superradiant instabilities, acting as sources of gravity and affecting the propagation of gravitational waves around the host black hole. During extreme-mass-ratio inspirals, the bosonic clouds will survive the inspiral phase and can affect the quasinormal-mode frequencies of the perturbed black-hole-bosonic-cloud system. In this work, we compute the shifts of gravitational quasinormal-mode frequencies of a rotating black hole due to the presence of a surrounding bosonic cloud. We then perform a mock analysis on simulated Laser Interferometer Space Antenna observational data containing injected ringdown signals from supermassive black holes with and without a bosonic cloud. We find that with less than an hour of observational data of the ringdown phase of nearby supermassive black holes such as Sagittarius A* and M32, we can rule out or confirm the existence of cloud-forming ultralight bosons of mass ∼10-17 eV .