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Many-body interacting systems cannot generally be treated with analytical tools alone, making numerical methods essential for studying strongly correlated phases. In fractional quantum Hall systems, conformal field theory correlators provide a way to construct exact matrix product state representations of model wavefunctions such as the Laughlin state. Fractional Chern insulators realize analogous physics in lattice systems, but the intrinsic lattice scale obstructs a direct application of this CFT-based construction. We show that ideal Chern bands provide a setting in which this obstruction can be overcome, and derive an exact MPS representation for Laughlin model states in a hybrid Wannier basis on the torus.