Nanotechnological applications have been proposed which require components to self-assemble into mesoscale structures. For, example, nanoscale sensors, quantum memory devices, or photonic materials might comprise regular arrays of particles assembled on a substrate. Previously, we studied the random sequential adsorption kinetics and the structural phase behavior of a two-dimensional model of particles (hard disks) tethered to a substrate. The tethers restrict particle surface mobility which affects the nonequilibrium phases of the developing monolayer adsorbed to the surface. Here, we explore the effect of Gaussian polydispersity on the tethered random sequential adsorption process. Liquid, hexatic, and crystal phases are observed in the simulations. For size-monodisperse systems, short tethers tone particle radius or less) allow only liquid structures, intermediate tethers tone to four particle radii) allow a hexatic structure at high coverages, and long-tethered systems develop through liquid and hexatic phases before becoming crystalline at high coverages. Polydispersity disrupts the order. Systems over approximately 8% polydispersity remain liquid, and systems between about 7 and 8% polydispersity form a hexatic phase even with very long tethers. For sufficiently long tethers, crystal formation requires 5-7% polydispersity or less. Histograms of particle size distributions on the surface reveal that short-tether systems yield bimodal distributions because of the persistence of small gaps in the layer. In contrast, systems with adequate surface mobility organize locally, which prevents small gaps and retains unimodal size distributions. Kinetics for polydisperse, tethered, random sequential adsorption processes follow a power law with constants that change rapidly for tether lengths less than one particle radius and polydispersities up to 10%. Jamming limit coverages generally increase with polydispersity but decrease as surface order is destroyed.