We introduce the deformable particle (DP) model for cells, foams, emulsions, and other soft particulate materials, which adds to the benefits and eliminates deficiencies of existing models. The DP model combines the ability to model individual soft particles with the shape-energy function of the vertex model, and adds arbitrary particle deformations. We focus on 2D deformable polygons with a shape-energy function that is minimized for area a_{0} and perimeter p_{0} and repulsive interparticle forces. We study the onset of jamming versus particle asphericity, A=p_{0}^{2}/4πa_{0}, and find that the packing fraction grows with A until reaching A^{*}=1.16 of the underlying Voronoi cells at confluence. We find that DP packings above and below A^{*} are solidlike, which helps explain the solid-to-fluid transition at A^{*} in the vertex model as a transition from tension- to compression-dominated regimes.