It is known in the Knights of the Old Republic series that vibroblades were made from a special material that couldn't be cut by lightsabers. Then, why aren't the clone troopers using any?
Close at hand sat a stubby treddroid tasked with monitoring the progress of a mining probe that was sampling a rich vein of cortosis ore at the bottom of a deep shaft. A fabled ore, some called it -- owing to its scarcity, but even more for its intrinsic ability to diminish the effectiveness of the Jedi lightsaber. For that reason, the Jedi Order had gone to great lengths to restrict mining and refinement of the ore. If not the bane of the Order’s existence, cortosis was a kind of irritant, a challenge to their weapon’s reputation for fearsome invincibility.
Legends novel Darth Plagueis, p. 10 (emphasis added)
Since cortosis was mostly unavailable, vibroblades were unsuitable as a weapon against a Jedi. Blasters aren't much better since Jedi can deflect blaster bolts, but at least they put some distance between the user and the Jedi's lightsaber (Jango Fett, for example, proved how effective blasters could be against a Jedi, and how important it was to maintain one's distance from a Jedi).
Finally, clone troopers and stormtroopers weren't intended to fight Jedi in most situations (especially clone troopers, who until Order 66 were allied with the Jedi). Consequently, there was little reason to equip them with vibroblades (especially since it would be far too expensive to equip an entire army with cortosis-weave vibroblades, given the scarcity of cortosis).
Well an out of universe reason is that the idea was not yet developed when the movies came out, but I hate out of universe answers.
In universe: Well the primary weapons of all clones is blasters, as they are fighting droids who are also using blasters mostly. A vibro blade would not be an effective weapon in that situation. If anyone should be using one it would be droids, but it was not really in the movies, with some exceptions:
-In the cartoon Clone Wars they added agile droids that used bladed sword like weapons which I assume are vibro blades.
-In episode two and three(possibly) there were droids wielding bowstaffs that were electrified on both ends. These were made of a material that could have been used for vibro blades.
-In some video games you could carry a small combat knife/vibro blade type weapon.
But then they attacked the Jedi, and in that situation vibro blades could perhaps be useful. Consider though troopers were not generally trained for vibro blade use so any 1v1 attack on a Jedi would fail. Blasters were fairly effective it seemed though, so vibro blades were not really necessary.
During the old republic vibro blades were very common. Due to the number of force users and lightsabers being used, a counter to lightsabers was produced. As the sith were defeated and the number of forcer users slowly diminished, weapons to counter force users were less in demand, leading to the disappearance of such weapons.
I am not saying vibro blade type weapons are not at all used around the time of movies, they are just uncommon and not shown in the movies.
In the Star Wars: The Clone Wars novel by Karen Traviss, vibroblades are used by the troopers of the 501st Legion during the Battle of Teth, though this is not shown during any of the TV show episodes or the movie.
Coric snatched the rifle from the crippled droid as it tottered backward; Nax grabbed a lump of masonry and battered another droid until its head caved in. Rex ejected the vibroblade from his forearm plate and jumped onto a droid, tipping it off balance and gouging out its photoreceptors. As it flailed blindly, he severed all the control cables to its head.
Perhaps other Clone Trooper units made use of these weapons.
According to Wookieepedia, Vibroblades continued to be of use from the Old Republic on. I can personally remember the lead miniatures, such as of the Gamorreans in front of Jabba's Palace, being called out as having vibroblades on their halberds, and the RPG sourcebooks include vibroblades as standard. What had changed by the time of the first two trilogies is that "the cortosis-weave became less common when the probability of fighting a lightsaber-wielding opponent decreased. By the time of the Galactic Civil War, knowledge of the cortosis-weave had faded, and the cortosis mineral itself had become exceedingly rare."
As to why the clone troopers didn't use any, presumably it was because they were primarily trained to use energy weapons as a formation rather than to engage in individual melee. As the fighting force moved to Stormtroopers, melee weapons would be further discouraged because, just as with our history, it's easier to teach someone to fire a gun than it is to teach them to be a competent melee fighter.
To give a real-world comparison, think about how many soldiers you see running around with swords. Not only do they have little practical use,but the effectiveness of the clone troopers would also be drastically reduced by the split time between swordplay training and conventional training. Overall, equipping the standard infantryman with vibroblades would not be worth the effort. Specialty units such as ARC troopers or assassins, on the other hand, would be worth the time.
Although I can't answer the question regarding Clone equipment, I can say that vibroblades existed in some quantity until at least 20 years after the destruction of the second Death Star.
From the new canon novel Bloodline, set ~20 years after Return of the Jedi, and just before the rise of the First Order:
In the stark-white training chamber, he remained stock-still, poised with the quarterstaff in his hand. His formfitting gray workout gear had been drenched with sweat for a long time now, but Ransolm remained determined to press on. Most people neglected training with non-powered weapons; these were the same people who got themselves killed because their blasters ran out of charge. He intended to be able to fight with whatever was at hand, whether as sophisticated as the latest model of vibroblade or as brutish as a long stick.
- Star Wars: Bloodline