  A search is presented for a Higgs boson that is produced via vector boson
fusion and that decays to an undetected particle and an isolated photon. The
search is performed by the CMS Collaboration at the LHC, using a data set
corresponding to an integrated luminosity of 130 fb$^{-1}$, recorded at a
center-of-mass energy of 13 TeV in 2016-2018. No significant excess of events
above the expectation from the standard model background is found. The results
are interpreted in the context of a theoretical model in which the undetected
particle is a massless dark photon. An upper limit is set on the product of the
cross section for production via vector boson fusion and the branching fraction
for such a Higgs boson decay, as a function of the Higgs boson mass. For a
Higgs boson mass of 125 GeV, assuming the standard model production rates, the
observed (expected) 95% confidence level upper limit on the branching fraction
is 3.5 (2.8)%. This is the first search for such decays in the vector boson
fusion channel. Combination with a previous search for Higgs bosons produced in
association with a Z boson results in an observed (expected) upper limit on the
branching fraction of 2.9 (2.1)% at 95% confidence level.
