Housing Opportunity Through Modernization Act of 2016 This bill amends the United States Housing Act of 1937 and other housing laws to modify the Department of Housing and Urban Development (HUD) rental assistance (including section 8 low-income [voucher]) and public housing programs, Federal Housing Administration (FHA) requirements for condominium mortgage insurance, and the Department of Agriculture (USDA) single family housing guaranteed loan program. The bill revises the requirements for Public House Agencies (PHAs) to inspect dwelling units before making housing assistance payments to ensure that units comply with housing quality standards. A PHA shall review the incomes of assisted families in dwelling units, and may not rent a dwelling unit to or assist families with net family assets exceeding $100,000 annually (adjusted for inflation) or an ownership interest in property that is suitable for occupancy. The bill also revises requirements for: PHA project-based assistance, including vouchers; the public notice requirements for proposed Fair Market Rents; and the Family Unification Program, especially coordination between PHAs and public child welfare agencies in carrying it out. The Housing Act of 1949 is amended to permit USDA to delegate to preferred lenders its loan approval authority for the Rural Housing Service single family housing guaranteed loan program. The National Housing Act is amended to require the FHA to modify its certification requirements for condominium mortgage insurance. The McKinney-Vento Homeless Assistance Act is amended to require HUD to define the ''geographic area'' for purposes of the Continuum of Care Program (which awards project sponsors or unified funding agencies competitive grants focused on addressing the long-term housing and services needs of homeless individuals and families). Local governments receiving Emergency Solutions Grants may distribute all or part of the assistance to PHAs or Local Redevelopment Authorities. (The grants are awarded to assist the homeless and prevent homelessness. Distribution of the grants is currently limited to nonprofit organizations.) The bill transfers to the Office of the Secretary the Special Assistant for Veterans Affairs, currently in the Office of the Deputy Assistant Secretary for Special Needs. HUD and the Department of Veterans Affairs must report annually to Congress on the number of veterans assisted by HUD programs, coordination of services for veterans, and the cost of administering programs to veterans. The bill also revises the formula and requirements for distributing funds under the Housing Opportunities for Persons With AIDS (HOPWA) Program.