What is a wavefunction, in the context of String Theory?

I think one important thing to mention is that Copenhagen Interpretation of Quantum Mechanics is not the only interpretation out there; in particular there is no "need" for wavefunction collapse if you don't want it. Alternative interpretations include the Many Worlds Interpretation (which makes sense in light of the development of Decoherence in quantum theories; for more on this interpretation I'd recommend this paper by Max Tegmark).

I don't think String Theory should be viewed as something that can resolve between different interpretations of Quantum Mechanics; after all String Theory is still a Quantum Theory. In particular, you still have wavefunctions as before; but instead of describing what you might think of as 'point particles', they describe strings.