I think that a good starting point is the paper by Ted Erler and his ex-adviser David Gross
and references within, and citations thereof. They write down bosonic open string field theory in such a way that it's manifestly local where the locus reduces to the interaction point.
A warning: this is just open string field theory so there are no gravitons in the manifest spectrum. However gravitons, much like all other closed strings, may be seen as poles in the open strings' scattering amplitudes.
The full theory including the physical gravitons probably can't be made fully local in the same sense because gravitational amplitudes only make sense off-shell, even though this statement could be modified in light-cone gauge, too. There is a problem with closed string field theory if it has to be covariant; but it works in the light cone gauge as well as open string field theory. I am not quite sure why Ted and David only considered open strings.
The explicit construction of theirs clarifies the reason why the high-energy behavior of the open string scattering amplitudes – with the angle scaling in a certain way with energy – saturates the inequality one obtains from locality. This is apparently only the case for perturbative open string theory; the full scattering amplitudes at strong coupling ultimately switch to the black hole regime which implies a faster decrease of the amplitudes with energy.
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