See the following papers, where it is shown that suitable spaces of invertible Fourier integral operators form an infinite dimensional Frechet Lie group with corresponding spaces of pseudifferential operators as Lie algebra. There are also papers by Omori, Maeda, etc with somewhat weaker results.
MR0863703 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf A Lie group structure for Fourier integral operators. Math. Ann. 276 (1986), no. 1, 19–41. (Reviewer: J. J. Duistermaat) 58G15 (22E65 58B25)
MR0826458 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf A Lie group structure for pseudodifferential operators. Math. Ann. 273 (1986), no. 4, 529–551. (Reviewer: J. J. Duistermaat) 58G15 (22E65 35S05 47G05 58B25)
MR0823314 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications. Infinite-dimensional groups with applications (Berkeley, Calif., 1984), 1–69, Math. Sci. Res. Inst. Publ., 4, Springer, New York, 1985. (Reviewer: J. J. Duistermaat) 58D05 (22E65 58B25 58F99 58G15)
MR0815694 Schmid, Rudolf; Adams, Malcolm; Ratiu, Tudor The group of Fourier integral operators as symmetry group. XIIIth international colloquium on group theoretical methods in physics (College Park, Md., 1984), 246–249, World Sci. Publishing, Singapore, 1984. 58G15 (22E65)
Added:
Pseudodifferential operators are not local operators in general. So it does not make sense to speak of sheaf properties in a naive sense. I have no idea what sheafification would lead to; I doubt, that it is anything useful or interesting. Microlocal analysis does look superficially like that, but not really.