We study a two-dimensional electron gas in the integer quantum Hall regime coupled to a single spatially modulated cavity mode in the static off-resonant limit. Retaining both paramagnetic and diamagnetic contributions on equal footing, we derive a gauge complete single-particle Hamiltonian in which virtual photon exchange generates a periodic kinetic renormalization and a diamagnetic scalar potential. Both terms transfer momentum by 2k₀ and displace guiding centers by Δ X=2k₀ℓ_B², forming a finite-k₀ momentum ladder. Diagonalizing the strip Hamiltonian, we find that the ladder reconstructs the Landau-level band structure and redistributes, rather than multiplies, the associated chiral edge states. Occupied-subspace Chern numbers remain C_ν=ν for ν=1,2,3 throughout 0≤λ_cav≤ 0.5, while the bulk gaps decrease without closing. Strip Kubo responses show cavity-induced displacements of the Hall-step energies; at finite width the plateau heights are noninteger and approach the integers as Lₓ increases. The ladder is spectrally resolved at k₀ℓ_B=0.5 and collapses at k₀ℓ_B=0.1. These results isolate a spatially local, finite-k₀ momentum ladder as a clean-system mechanism by which a structured vacuum field reconstructs quantum Hall spectra and strip transport without changing the integer topology, complementary to cavity-mediated all-to-all hopping.