Planning for Aging Veterans Act of 2021 This bill addresses the long-term care of veterans, including by implementing policies for the administration of state homes. First, the bill requires the Department of Veterans Affairs (VA) to develop a strategy for the long-term care of veterans. The VA must develop a standardized process for entering into sharing agreements between state homes and VA medical centers. A state home is a home established by a state for veterans who are disabled by age, disease, or otherwise and incapable of earning a living because of such disability. The term also includes a home that furnishes nursing home care for veterans. Under the bill, the VA must furnish prescribed drugs and medicines to veterans who are catastrophically disabled and for whom the VA is paying a per diem for nursing home or domiciliary care in a state home. Generally, veterans are catastrophically disabled if they have a permanent severely disabling injury, disorder, or disease that compromises the ability to carry out the activities of daily living. The VA must monitor any contractor it uses to conduct inspections of state homes, including by reviewing inspections, reporting deficiencies, and publishing inspection results and any planned corrective actions on a public VA website. The VA must implement a two-year pilot program to provide geriatric psychiatry assistance to eligible veterans at state homes. Finally, the VA must work with public housing authorities and local organizations to assist aging homeless veterans in accessing existing housing and supportive services.