A search for anomalous electroweak production of WW, WZ, and ZZ boson pairs in association with two jets in proton-proton collisions at $\sqrt{s}=13$ TeV at the LHC is reported. The data sample corresponds to an integrated luminosity of $35.9$ fb$^{-1}$ collected with the CMS detector. Events are selected by requiring two jets with large rapidity separation and invariant mass, one or two leptons (electrons or muons), and a W or Z boson decaying hadronically. No excess of events with respect to the standard model background predictions is observed and constraints on the structure of quartic vector boson interactions in the framework of dimension-8 effective field theory operators are reported. Stringent limits on parameters of the effective field theory operators are obtained. The observed $95\%$ confidence level limits for the S0, M0, and T0 operators are $-2.7<f_{\text{S0}}/ \Lambda^{4}<2.7$, $-1.0<f_{\text{M0}}/ \Lambda^{4}<1.0$, and $-0.17<f_{\text{T0}}/ \Lambda^{4}<0.16$, in units of TeV{}$^{-4}$. Constraints are also reported on the product of the cross section and branching fraction for vector boson fusion production of charged Higgs bosons as a function of mass from 600 to 2000 GeV. The results are interpreted in the context of the Georgi--Machacek model.