Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian?
If not, is this related to many vacua existing?
This is not a bad question. I'd appreciate it if you expanded some to give context and explain what you're thinking about.
Short answer: No, there isn't. It's not clear that M-theory has a description in the Lagrangian framework.
Longer answer: The nature of spacetime in M-theory is radically different from the nature of spacetime in classical & quantum field theory. Exactly how different is still under investigation. But it is not clear that spacetime is infinitely divisible in this theory. If short distances don't exist, then it is not clear we should be using the Lagrangian framework to describe fundamentla physics, since these implicitly associate degrees of freedom to all distance scales.
That said: Lagrangians do play a role in the physics of M-theory. They are used to describe worldvolume QFTs, which are effective QFTs which describe how strings and branes see the classical spacetime around them: The matrix model of D0-branes. The D3-branes of AdS/CFT are described by N=4 gauge theories. The basis of perturbative string theory is a nonlinear sigma model. A stack of 5-branes in M-theory is described by a (2,0) theory.
There is a candidate for M-theory lagrangian for coincident M2 branes. This lagrangian may provide the first M –theory lagrangian describing the quantum dynamics of membranes. http://arxiv.org/abs/hep-th/0611108