You are correct in that sound waves would carry momentum. However, there are three reasons why it's not relevant for the results of the inelastic collision.
The first is that the inelastic collision is a theoretical model and therefore happens in the famous frictionless vacuum. In fact, the model doesn't include any consideration about where the energy goes which disappears during the collision. It's simply assumed to be lost. A friend of mine once did the force integration approach of modelling a collision and could show that an inelastic collision is equal to assuming that at the point where the two centers of mass are nearest the elastic force stops working.
So where does the energy go? This leads us to our second reason. Where exactly it goes is a complicated solid state physics rabbit hole, but basically, it at first plastically deforms the object (by overcoming the attraction between its constituent molecules) and from there decays into heat. You can have other channels, like the sound you mentioned, but all of these can be assumed for the purpose of the excercise to be isotropic, meaning they give off the same energy in all directions. This means they are momentum-free, because the momentum carried by the sound wave travelling in one direction is exactly cancelled by the momentum of the sound wave travelling in the opposite direction. In total, no momentum is carried, and thus we can assume the inelastic collision to preserve momentum on its macroscopic scale.
And thirdly, even if our sound is not isotropic, and therefore carries some momentum, as momentum is proportional to mass, and the mass of the air carrying the sound is consideraly smaller than that of the two colliding objects (and their speed not enough orders of magnitude larger to cancel that), the momentum loss would be below 1% of the total, which is negilible.
Additionally, both the elastic and inelastic collision are theoretical models meant to show students the possible spectrum of results of two objects hitting each other. In reality, no collision is purely elastic or inelastic, but always somewhere in between. They are taught not because they're perfect models of reality, but because their assumptions makes the math really easy. You can calculate more complicated collisions (even including parts flying off and carrying momentum), but that math gets horrible quickly.