  We introduce a general method for constructing modules for $0$-Hecke algebras
and supermodules for $0$-Hecke-Clifford algebras from diagrams of boxes in the
plane, and give formulas for the images of these modules in the algebras of
quasisymmetric functions and peak functions under the relevant characteristic
map. As initial applications, we resolve a question of Jing and Li (2015),
introduce a new basis of the peak algebra analogous to the quasisymmetric Schur
functions, uncover a new connection between Schur $Q$-functions and
quasisymmetric Schur functions, give a representation-theoretic interpretation
of families of tableaux used in constructing certain functions in the peak
algebra, and establish a common framework for known $0$-Hecke module
interpretations of bases of quasisymmetric functions.
