String theory as we know it admits only S-matrix as an observable. By its definition an S-matrix is a non-local object, it tells you about transition amplitudes between asymptotic states in past and future infinity. You cannot even ask local questions in spacetime, unless you somehow extend the formalism (which is the goal of string field theory, more about this below).
This is not (in my opinion) a quirk of the formalism. String theory is a quantum theory of gravity, and at long distances it coincides with General Relativity. GR also does not allow for local observables. Mathematically it is because there are no local diffeomorphism invariant quantities. Physically it is because there is no systematic way to locally probe the system without disturbing it — to construct a localized classical probe (a.k.a. measuring device) you’d want it to be very massive (i.e. has many degrees of freedom) to suppress quantum fluctuations. Alas, if it couples to gravity it then back-reacts on the geometry. If gravity couples weakly you can construct approximately localized probes, but this does not work in the fully quantum gravitational regime.
All of this does not mean the theory is not local, just that you have to be careful how to phrase the question and make sure it makes sense. There are a few indications that if you ask the question the right way string theory is local in some sense. Two such indications that come to mind:
For a local QFT, the S-matrix obeys certain properties which follow from locality. Turns out string theory obeys those as well. This of course does not, strictly speaking, imply that string theory is local, but it is an indication that it is not obviously non-local.
In extension of the formalism, like open SFT, in some sense interactions are local on the worldsheet - strings interact only when they touch in spacetime. Opinions vary on what this means, FWIW in my mind SFT is inherently perturbative, and for perturbative gravity perhaps it is not surprising one can construct quasi-localized probes. In any event, this is not (I don't think) a gauge invariant statement, so it cannot be made as sharp as one would have liked.
As for your specific question: In perturbative string theory, only after integrating over conformal structures you have a chance of getting objects which make sense physically, which are the S-matrix elements. If you fix the conformal structure you get objects which cannot be interpreted physically (e.g. they have "ghosts", negative norm states in the Hilbert space).
Specifically: probes of the theory are realized as punctures on the worldsheet, and conformal invariance (achieved by integration over conformal structures) pushes their location in spacetime to asymptotic null infinity. Heuristically, this is because a puncture is conformal to an infinitely long tube emanating from the worldsheet . Less heuristically, conformal invariance on the vertex operator inserted at the puncture (which expresses the specific probe of the theory) translates to mass-shell conditions in spacetime (which is the Fourier transform of the previous statement). Since you don't have all the Fourier modes of your probe, there is no way to localize it in spacetime.
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