  We apply the worldline formalism to amplitudes in scalar quantum
electrodynamics (QED) involving open scalar lines, with an emphasis on their
non-perturbative gauge dependence. At the tree-level, we study the scalar
propagator interacting with any number of photons in configuration space as
well as in momentum space. At one-loop we rederive, in an efficient way, the
off-shell vertex in an arbitrary dimension and any covariant gauge.
Generalizing the Landau-Khalatnikov-Fradkin transformation (LKFT) for the
non-perturbative propagator, we find simple non-perturbative transformation
rules for arbitrary x-space amplitudes under a change of the covariant gauge
parameter in terms of conformal cross ratios.
