  In order to reach (sub-)per cent level precision in lattice calculations of
the hadronic vacuum polarisation, isospin breaking corrections must be
included. This requires introducing QED on the lattice, and the associated
finite-size effects are potentially large due to the absence of a mass gap.
This means that the finite-size effects scale as an inverse polynomial in $L$
rather than being exponentially suppressed. Considering the
$\mathcal{O}(\alpha)$ corrected hadronic vacuum polarisation in
QED$_{\mathrm{L}}$ with scalar QED as an effective theory, we show that the
first possible term, which is of order $1/L^{2}$, vanishes identically so that
the finite-size effects start at order $1/L^{3}$. This cancellation is
understood from the neutrality of the currents involved, and we show that this
cancellation is universal by also including form factors for the pions. We find
good numerical agreement with lattice perturbation theory calculations, as well
as, up to exponentially suppressed terms, scalar QED lattice simulations.
