There are two main classes of physics-based models for two-dimensional cellular materials: packings of repulsive disks and the vertex model. These models have several disadvantages. For example, disk interactions are typically a function of particle overlap, yet the model assumes that the disks remain circular during overlap. The shapes of the cells can vary in the vertex model, however, the packing fraction is fixed at $\phi=1$. Here, we describe the deformable particle model (DPM), where each particle is a polygon composed of a large number of vertices. The total energy includes three terms: two quadratic terms to penalize deviations from the preferred particle area $a_0$ and perimeter $p_0$ and a repulsive interaction between DPM polygons that penalizes overlaps. We performed simulations to study the onset of jamming in packings of DPM polygons as a function of asphericity, ${\cal A} = p_0^2/4\pi a_0$. We show that the packing fraction at jamming onset $\phi_J({\cal A})$ grows with increasing ${\cal A}$, reaching confluence at ${\cal A} \approx 1.16$. ${\cal A}^*$ corresponds to the value at which DPM polygons completely fill the cells obtained from a surface-Voronoi tessellation. Further, we show that DPM polygons develop invaginations for ${\cal A} > {\cal A}^*$ with excess perimeter that grows linearly with ${\cal A}-{\cal A}^*$. We confirm that packings of DPM polygons are solid-like over the full range of ${\cal A}$ by showing that the shear modulus is nonzero.