My take on it (using some EU): If you look at Yoda in the PT he still seems somewhat fragile. Walking slowly with a cane until he calls upon the force to fight with saber. We don't see him do that in OT. There doesn't seem any reason for him to do so. He probably doesn't have a saber but it's not impossible that he does. He would require the crystals available at three different planets as well as parts to build one. It's possible that since Obi-wan either had or was able to procure some that he was too (Luke was able to build his using what he found at Obi-wan's old hutt). Palpatine does not care for lightsabers. He feels that they are not as a formidable a weapon as the power of the Dark Side. He didn't appear very comfortable using one in RotS (although this was partly due to Ian not being very comfortable) and in the end always relied on his force lightning. He felt that the Sith had outgrown lightsabers and only used them to humilate the Jedi. That's why he called Luke's lightsaber a Jedi's weapon.