Measuring the non-Gaussianity of the initial matter density fluctuations may provide powerful insights into cosmic inflation and the origin of structures. Although current information on primordial non-Gaussianity comes mostly from the cosmic microwave background, obtaining constraints from lower redshift observables will be an important task for future surveys. We study the ability of future weak lensing (WL) surveys to constrain primordial non-Gaussianity of the local type. We use a large ensemble of simulated WL maps with survey specifications relevant to Euclid and Large Synoptic Survey Telescope. The simulations assume cold dark matter cosmologies that vary certain parameters around fiducial values: the non-Gaussianity parameter fNL; the matter density parameter Ωm; the amplitude of the matter power spectrum σ8; the spectral index of the primordial power spectrum ns and the dark-energy equation-of-state parameter w0. We assess the sensitivity of the cosmic shear correlation functions, the third-order aperture mass statistics and the abundance of shear peaks to these parameters. We find that each of the considered probes provides unmarginalized constraints of Δ fNL ˜ 20 on fNL. Marginalized constraints from any individual WL probe are much weaker due to strong correlations between parameters. However, the parameter errors can be substantially reduced by combining information from different WL probes. Combining all WL probes yields the following marginal (68 per cent confidence level) uncertainties: Δ fNL ˜ 50; Δ Ωm ˜ 0.002; Δσ8 ˜ 0.004; Δ ns ˜ 0.007 and Δw0 ˜ 0.03. We examine the bias induced by neglecting fNL on the constraints on the other parameters. We find σ8 and w0 to be the most affected. Moreover, neglecting non-Gaussianity leads to a severe underestimation of the uncertainties in the other cosmological parameters. We conclude that a full exploitation of future WL surveys requires a joint analysis of different WL probes. Furthermore, if not taken into account, a non-vanishing level of primordial non-Gaussianity will bias the estimated cosmological parameters and uncertainties for future surveys.