  Extreme mass-ratio inspirals are crucial sources for future space-based
gravitational wave detections. Gravitational waveforms emitted by extreme
mass-ratio inspirals are closely related to the orbital dynamics of small
celestial objects, which vary with the underlying spacetime geometry. Despite
the tremendous success of general relativity, there are unsolved issues such as
singularities in both black holes and cosmology. Loop quantum gravity, a theory
addressing these singularity problems, offers a framework for regular black
holes. In this paper, we focus on periodic orbits of a small celestial object
around a supermassive quantum-corrected black hole in loop quantum gravity and
compute the corresponding gravitational waveforms. We view the small celestial
object as a massive test particle and obtain its four-velocity and effective
potential. Our results indicate that the quantum parameter $\hat{\alpha}$
influences the shape of the effective potential. We explore the effects of
quantum corrections on marginally bound orbits, innermost stable circular
orbits, and other periodic orbits. Using the numerical kludge scheme, we
further explore the gravitational waveforms of the small celestial object along
different periodic orbits. The waveforms exhibit distinct zoom and whirl phases
in a complete orbital period, closely tied to the quantum parameter
$\hat{\alpha}$. We also perform a spectral analysis of the gravitational waves
from these periodic orbits and assess their detectability. With the steady
progress of space-based gravitational wave detection programs, our findings
will contribute to utilizing extreme mass-ratio inspirals to test and
understand the properties of quantum-corrected black holes.
