The first measurement of the dependence of γγ→μ^{+}μ^{-} production on the multiplicity of neutrons emitted very close to the beam direction in ultraperipheral heavy ion collisions is reported. Data for lead-lead interactions at sqrt[s_{NN}]=5.02 TeV, with an integrated luminosity of approximately 1.5 nb^{-1}, are collected using the CMS detector at the LHC. The azimuthal correlations between the two muons in the invariant mass region 8<m_{μμ}<60 GeV are extracted for events including 0, 1, or at least 2 neutrons detected in the forward pseudorapidity range |η|>8.3. The back-to-back correlation structure from leading-order photon-photon scattering is found to be significantly broader for events with a larger number of emitted neutrons from each nucleus, corresponding to interactions with a smaller impact parameter. This observation provides a data-driven demonstration that the average transverse momentum of photons emitted from relativistic heavy ions has an impact parameter dependence. These results provide new constraints on models of photon-induced interactions in ultraperipheral collisions. They also provide a baseline to search for possible final-state effects on lepton pairs caused by traversing a quark-gluon plasma produced in hadronic heavy ion collisions.