Remotely plausible, but in almost no way effective. The only bullet that could be made from obsidian would be one for a smoothbore gun, or a sub-caliber projectile (think shot or canister). Rifle bullets require a degree of compressibility to fit the grooves, and obsidian is practically incompressible. Trying to fire it through a rifle will, depending on the specifics of the situation, either fragment your bullet, explode your gun, or deform your barrel enough for the blast to vent.
The other problen is that obsidian is brittle, hard and sharp. Shaping it is possible, but shaping it into either a smooth ball or conical cylinder is going to be extremely laboorious. With modern power tools, any skilled machinist or stonemason can do it, of course, but when all you have is rocks, sand, and bone or antler tools, it will be a sisyphean task even with a primitive lathe or ball mill.
Now you have your obsidian ball and musket to fire it from, you get problem three. Obsidian is considerably lighter than lead, and hold less kinetic energy. Its armour-penetrating qualities will thus be limited. Against any organic armours (fabric, leather, wood), a lead bullet is likely to fare as well anway, since the difference in hardness is not that great. Against a metal armour, the chance is good that an obsidian bullet will shatter (remember - brittle). If it doesn't, it will likely do more damage than a lead one of equal weight and velocity, but I doubt the tradeoff is worth it.