Context. Gamma-ray emission at energies ≥100 MeV has been detected from nine novae using the Fermi Large Area Telescope (LAT), and can be explained by particle acceleration at shocks in these systems. Eight out of these nine objects are classical novae in which interaction of the ejecta with a tenuous circumbinary material is not expected to generate detectable gamma-ray emission. Aim. We examine whether particle acceleration at internal shocks can account for the gamma-ray emission from these novae. The shocks result from the interaction of a fast wind radiatively-driven by nuclear burning on the white dwarf with material ejected in the initial runaway stage of the nova outburst. Methods. We present a one-dimensional model for the dynamics of a forward and reverse shock system in a nova ejecta, and for the associated time-dependent particle acceleration and high-energy gamma-ray emission. Non-thermal proton and electron spectra are calculated by solving a time-dependent transport equation for particle injection, acceleration, losses, and escape from the shock region. The predicted emission is compared to LAT observations of V407 Cyg, V1324 Sco, V959 Mon, V339 Del, V1369 Cen, and V5668 Sgr. Results. The ≥100 MeV gamma-ray emission arises predominantly from particles accelerated up to ~100 GeV at the reverse shock and undergoing hadronic interactions in the dense cooling layer downstream of the shock. The emission rises within days after the onset of the wind, quickly reaches a maximum, and its subsequent decrease reflects mostly the time evolution of the wind properties. Comparison to gamma-ray data points to a typical scenario where an ejecta of mass 10⁻⁵–10⁻⁴ M_⊙ expands in a homologous way with a maximum velocity of 1000–2000 km s⁻¹, followed within a day by a wind with a velocity <2000 km s⁻¹ and a mass-loss rate of 10⁻⁴–10⁻³ M_⊙ yr⁻¹ declining over a time scale of a few days. Because of the large uncertainties in the measurements, many parameters of the problem are degenerate and/or poorly constrained except for the wind velocity, the relatively low values of which result in the majority of best-fit models having gamma-ray spectra with a high-energy turnover below ~10 GeV. Our typical model is able to account for the main features in the observations of the recent gamma-ray nova ASASSN-16ma. Conclusions. The internal shock model can account for the gamma-ray emission of the novae detected by Fermi LAT. Gamma-ray observations hold potential for probing the mechanism of mass ejection in novae, but should be combined to diagnostics of the thermal emission at lower energies to be more constraining.