I was wondering if it is possible for an object of a certain material to be put in a buoyant medium in such a way that the buoyant force (caused by the pressure difference “above” and “below” said object and by the interactions with molecules of the medium under that regime) creates an acceleration that causes a motion that exceeds the sound speed of the material of the object.
I cannot think of scenario where this is possible (though that's not saying much) outside of a plasma. However, assume it were possible to accelerate, say, a balloon fast enough to exceed the speed of sound of the gas within the balloon. What then? Is that along the lines of your question?
If so, then let's assume the balloon were made of impermeable material (don't want it rupturing). In this scenario, the result would be one side of the balloon acting like a piston to generate a shock wave. The shock would technically be between those reflected off the balloon wall and the still incident particles.
Since this movement “ripples” through the object as it snakes upwards to get a new position in the medium, there is a limit as to how fast this type of slinky movement can happen, right?
This is a different question. The "ripples" to which you refer would be in the balloon material, not the medium contained within. I use the balloon example because the sound speed in nearly all solids is so high as to be physically restrictive for the thought experiment. These "ripples" would propagate at the speed of sound in the balloon material (e.g., some type of latex polymer or rubber), which is at least an order of magnitude higher than the speed of sound in the gas contained within (unless the whole balloon is crushed such that the pressure within is extremely large).
I am wondering what happens to the movement of the object as it approaches or exceeds this limit and if some aspects of buoyancy (perhaps dynamic friction) would make this construction impossible.
I think there are several other issues rendering this idea impossible (or just not feasibly testable). For instance, suppose we put our indestructable balloon at the bottom of the Mariana's trench and let it go. The buoyant force would be huge and the balloon would accelerate quickly, but fluid drag would prevent the speed from getting too large. That is, the terminal speed is limited by the drag on the accelerating and expanding balloon (cross section increases as it rises in, say, water).