It's possible to get a Wobbuffet with different moves through evolving a Wynaut, but there are only three different moves available though this, two of which are pretty useless. The moves are Charm, Encore and Splash. Encore definitely has some use, but the other two do not interest Wobbuffet at all. It also learns Safeguard through TM, but that's equally as useless.
However, this does not mean Wobbuffet is useless. The moves Wobbuffet learns through level-up actually got it banned from many tier in competitive battles in some of the past Generations! The reason for this is Wobbuffet's extremely high HP paired with the moves Counter and Mirror Coat. This is all Wobbuffet needs to essentially guarantee at least 1 KO per match given that you can predict the opponent's move. With Counter, if are hit by a physical move, it deals double the amount of damage the attack did to you back to the opponent. Same with Mirror Coat, except with special moves.
To give you an idea of how this is helpful, let's say you have a Focus Sash Wobbuffet at full health against a Heracross that you predict is about to use Megahorn. If you use Counter, and Wobbuffet has its highest possible HP stat (584), and the Heracross does use Megahorn, you would return a huge 1166 damage back (assuming the attack would have knocked you out)! This is because the attack left Wobbuffet on 1/584 HP thanks to its Focus Sash. 1166 is well over enough to KO any Pokemon in the game immediately, since the highest possible amount of HP is Blissey's 714.
And if you think the attack won't do more than half your HP in damage, you can use Encore to make it keep using the same move, which you can then use as Counter/Mirror Coat damage. Shadow Tag, Wobbuffet's ability, will keep them in the battle, essentially guaranteeing you the KO once again. And if Wobbuffet ends up weakened and you think it might die the next turn, you can use Destiny Bond, another one of its level-up moves, to bring down your opponent with you.
So yeah, Wobbuffet isn't entirely useless. But it is still a huge gimmick, and it can be stopped easily; status like poison from Toxic wrecks it, and Taunt pretty much stops it from doing anything.