  Models with vector-like leptons can strongly modify the lepton mass
generation mechanism and lead to correlated effects in lepton-Higgs couplings
and lepton dipole moments. Here we begin an analysis of higher-order
corrections in such models by setting up a renormalization scheme with full
on-shell conditions on the lepton self energies, masses and fields. A minimal
set of fundamental parameters is renormalized in the $\overline{\text{MS}}$
scheme. We provide a detailed discussion of lepton mixing and redundancies at
higher orders, show how the relevant counterterms can be obtained from the
renormalization conditions, and determine the $\beta$-functions corresponding
to the scheme. As a first application we calculate the one-loop effective
muon--Higgs coupling and analyse its correlation with the muon anomalous
magnetic moment $\Delta a_\mu^{\text{VLL}}$. In the interesting case of large
masses and opposite-sign coupling, the lowest-order correlation implies a fixed
value of $\Delta a_\mu^{\text{VLL}}$ around $22.5\times 10^{-10}$, while the
higher-order corrections significantly reduce this value to the interval
$(10...18)\times 10^{-10}$.
