Microbes across species and environments form biofilms, living materials composed of cells and extracellular polymers. Biofilm-dwelling cells benefit from emergent soft matter physics that sculpts three-dimensional morphologies and osmotically absorbs nutrients. Although biofilms are modeled as viscoelastic gels, the physical origins of the phase transition underlying their conversion from groups of cells to living gels have not been systematically investigated. Here, we show that Bacillus subtilis biofilms use polymer composition to tune their physical properties and drive gel formation. Using imaging and water immersion experiments with matrix knockout strains, we demonstrate the complementary roles of two polymers in this developmental transition: hydrophilic poly-{gamma}-glutamate swells colonies by absorbing water and exopolysaccharides serve as effective cross-linkers, causing a sol-gel-like phase transition that imparts structural integrity. With matrix knockout co-culture biofilms, we independently modulate the production of each polymer and reveal a phase space of biofilm morphologies. Colonies that produce both polymers develop macroscopic wrinkles. A thin-film model predicts biofilm wrinkling from swelling-generated internal strain coupled to elasticity. The model reproduces the shape of our observed morphological phase diagram. Our results demonstrate that bacteria leverage gelation to vary their material properties and morphologies, with implications for microbial ecology and engineering living matter.