Lightsabers go missing in Star Wars surprisingly often, despite their singular nature and importance to their wielder. With combat often taking place at high altitudes and limb loss a common occurrence, the occasional loss of the weapon is to be expected. Jedi and Sith alike seem adept at replacing them, though how formal or involved those replacements are depends on the time and circumstances.
One figure who seems to have taken that lesson to heart is Emperor Palpatine, who manages to lose his lightsaber at a key point in the course of the saga. As the Sith Lord who destroyed the Galactic Republic and wiped out the Jedi, however, he’s not one to leave anything to chance. Despite being unable to keep his hands on his weapon, he’s ready to replace it quickly, as well as possessing powers that can make up the difference even if he isn’t.
Palpatine doesn’t use his lightsabers very often, nor does he need to. With the Jedi Order intact and the Sith bound by the Rule of Two, he has to move in secret, and given his prominence in the Senate and galactic affairs in general, flashing a red lightsaber is apt to draw unwanted attention. Afterwards of course, he holds absolute power over the state and wouldn’t need to. If all else fails, he has his Force lightning, which can readily counter a Jedi’s weapon and even overwhelm it in some cases.
A yet canonically, he possessed not just one lightsaber, but a pair. Ultimate Star Wars, a canon reference book published in 2015, has the details, stating that he constructed both weapons during his apprenticeship to Darth Plagueis. Though unconfirmed, that makes it likely that he killed Plagueis with at least one of them.
He only wielded them twice, canonically, during the events of the Clone Wars. Both times, he did so as Darth Sidious with his public persona well hidden. The first took place during Star Wars: The Clone Wars, Season 5, Episode 16, “The Lawless,” when he confronted Darth Maul and his brother Savage Opress. The second took place in a canon comic, Dark Horse’s Darth Maul, Son of Dathomir, Part Four, in which the Emperor faces and ultimately defeats Mother Talzin of the Night Sisters. (The comic was based on a script from the unproduced eighth season of The Clone Wars and thus considered canon.) Both times he used a pair of lightsabers in securing his victory.
That helps explain the swift “reappearance” of his lightsaber in Star Wars: Episode III – Revenge of the Sith. Mace Windu disarms one during their battle in Palpatine’s office. But a short time later, Palpatine is armed with a second lightsaber during his battle with Yoda in the Senate chambers. He slips one out of his sleeve during the battle with Windu, which ends when the Jedi delivers a forceful kick and sends it flying. That leaves the second for Yoda’s fight, and Ultimate Star Wars has an answer for that, too. He kept one in a hidden compartment in his office, and – having lost the first during the fight with Windu -- presumably retrieved it immediately thereafter. He produced it from his sleeve again to fight Yoda, presumably a habit he always practiced during his double life.
Afterwards he had no need of a lightsaber, content to use his Force lightning in the rare moments he actually felt threatened. He was last seen wielding a lightsaber in the canon novel Lords of the Sith, and presumably still had it when he met his end in Star Wars: Episode VI – Return of the Jedi. But the care with which his lightsabers were presented – specifically in The Clone Wars, which retroactively dealt with the events of Revenge of the Sith – speaks to their importance in the saga. Losing one is more than a continuity error, something the franchise is well aware of, and knows how to address with seemingly incidental details.