Whightman axioms and locality of nets of observables are concepts that apply to spacetime local field theory. There is also another axiomatization of (perturbative) field theory, namely the S-matrix approach.
Historically, this was suggested in the 1960s as a fundamental axiomatization of QFT. Then the success of the quark model in QCD made spacetime local field theory become the popular paradigm that it is today. But in fact these days people in QFT are being attracted back to the S-matrix approach (this is what the "amplituhedron" story in super Yang-Mills theory is about).
In any case, the key fact to be aware of is that perturbative string theory (which, as opposed to its would-be non-perturbative version is a mathematically well-defined concept), is an example of an S-matrix theory, not of a spacetime local field theory. For entry points to this fact see for instance the string theory FAQ right at the beginning (also search the text for further occurences of "S-matrix").
So your question really is: How is causality encoded in S-matrix theories?
I am not in position to give an exhaustive reply to this question, but a) broadly causality is encoded in the analycity of the S-matrix and its singularity stucture and b) there is an extensive literature on this.
For what it's worth, already the Wikipedia page on S-matrix theory informs about analycity prinicples of the S-matrix and that they encode causality (see the paragraph Basic principles):
These principles [analycity of the S-matrix] were to replace the notion of microscopic causality in field theory,