The multiwavelength spectral and temporal variability observed in blazars set tight constraints on current theoretical emission models. Here, we investigate the relativistic magnetic reconnection process as a source of blazar emission in which quasi-spherical plasmoids, containing relativistic particles and magnetic fields, are associated with the emission sites in blazar jets. By coupling recent two-dimensional particle-in-cell simulations of relativistic reconnection with a time-dependent radiative transfer code, we compute the non-thermal emission from a chain of plasmoids formed during a reconnection event. The derived photon spectra display characteristic features observed in both BL Lac sources and flat spectrum radio quasars, with the distinction made by varying the strength of the external photon fields, the jet magnetization, and the number of pairs per proton contained within. Light curves produced from reconnection events are composed of many fast and powerful flares that appear on excess of a slower evolving envelope produced by the cumulative emission of medium-sized plasmoids. The observed variability is highly dependent upon the orientation of the reconnection layer with respect to the blazar jet axis and to the observer. Our model provides a physically motivated framework for explaining the multitime-scale blazar variability across the entire electromagnetic spectrum.