The higher-dimensional objects are the "branes", which were first found as extended black holes in supergravity. In M-theory, the fundamental objects are the 2-brane and the 5-brane, and strings arise from compactifying a brane, e.g. a 2-brane is wrapped around the eleventh dimension, and shows up as a string in the 10-dimensional limit.
The classical vibrations of a string can be described e.g. by Fourier analysis. For the quantum vibrations of a string, you treat the Fourier modes as quantum harmonic oscillators, and get the basic string spectrum. If the string is interacting, then it can emit and absorb new strings too.
A similar analysis can be performed for branes, but it is much harder; it has been the work of many years and is still unfinished. The classical vibrations of a brane are straightforward enough, it's just like the sheet of a drum. The quantum vibrations of a brane are described by a "worldvolume theory" whose degrees of freedom include the position of the brane in the external space - entirely analogous to the scalars in the "worldsheet theory" of a string, which correspond to the position vectors of the points along the string.
But just as a string can emit and absorb other strings, a brane can emit and absorb strings and other branes. More precisely, some strings and branes will be attached - the open strings attached to a D-brane, or the "open M2-brane" of M-theory, which is like a cylinder with the circular ends attached to the parent brane - and they may detach and become closed strings/branes, or while attached they may emit and absorb closed strings/branes.
Another phenomenon is the brane stack, where N branes are on top of each other. In the case of open strings, this means that each string stretches from brane "i" to brane "j" for some i,j <= N. The interactions of these strings are described by a gauge theory of rank N, such as U(N).
AdS/CFT at strong coupling is also relevant here, though I have never quite got the details straight. It appears to be like this: the CFT is the worldvolume theory of the black brane stack, and "string theory in AdS space" is what happens behind the event horizon of the stack. Another thing I'm not clear on is the extent to which every string and brane can be viewed as a sort of black hole or black hole remnant.
The bottom line is that the vibrations of higher-dimensional objects are certainly a part of string theory (e.g.), but the basic theory of branes is still being worked out, and the more quantum it gets, the less it looks like a simple "blob that vibrates".
Many people like to say that the ultimate form of string theory will not even feature space and time as fundamental, they will instead be "emergent" from something else. If that happens, what we now call the vibration of a string or a brane will also have to be "something else" fundamentally, though I don't know what. Or perhaps it won't even be part of the simplest formulation - perhaps stringy vibration is a sort of dynamical gauge freedom, a manifestation of a redundancy that is not strictly necessary.
