Jupiter has a sidereal rotation period or day of 9.925 hours.
The two innermost moons of Jupiter, Metis and Adrastea, have orbital periods of 7 hours 10 minutes 16 seconds and 7 hours 15 minutes 21 seconds respectively.
Saturn has a sidereal rotation period or day of 10 hours 33 minutes 38 seconds, or 0.4400231 Earth days.
The innermost confirmed moon, S/2009 S1, has an orbital period of 0.47 Earth days, which is longer than Saturn's day.
So I have checked the moons of two of the giant planets, leaving two giant planets, Uranus and Neptune, to be checked.
I note that a number of objects in the solar system which are not classified as planets, but as dwarf planets, Trans Neptunian Objects, asteroids, etc., have been discovered to have moons. If you are interested whether any of them orbit in less than a day of their primary, you should know that:
Of the objects within our Solar System known to have natural satellites, there are 76 in the asteroid belt (five with two each), four Jupiter trojans, 39 near-Earth objects (two with two satellites each), and 14 Mars-crossers.2 There are also 84 known natural satellites of trans-Neptunian objects.2 Some 150 additional small bodies have been observed within the rings of Saturn, but only a few were tracked long enough to establish orbits.
So there would be a lot of checking to do to find which of them might orbit faster than one day of their primary.