The theory of cosmic inflation (with the inflaton field) is an example of an effective quantum field theory - much like the Standard Model of particle physics (with quarks, leptons, photons, gauge bosons, etc.). And that's why, if it is correct, it must emerge as an approximation of stringy physics.
So inflation and string theory are compatible but string theory is much more general a theory, and the inflaton - if it exists - is just one of the many quantum fields predicted by string theory at lower energies (although the inflation scale is probably not too much lower than the string scale).
The most "mainstream" incorporation of inflation within string theory is the KKLMMT paper,
which has 600+ citations now. Of course, with letters such as KKLT in the list of authors, it's not shocking that those people rely on some anthropic tricks to get the slow-roll potential and other things. There may exist better models that predict all the required parameters more automatically. But they won't change the fact that inflation is just a derived concept from string theory.
There exist papers that try to geometrize the inflaton in much more stringy terms - it may be e.g. the distance between a brane and an antibrane that may annihilate. The resulting physics is more or less effectively equivalent to the inflation - it just gives an extra answer to the question "what the inflaton is made out of".