Handicap is a noun referring to (1) an advantage or penalty imposed to make a race or contest fair, (2) a hindrance, and (3) a physical or mental disability. It’s also a verb meaning (1) to assign handicaps, and (2) to impede. The adjective commonly used to describe people and groups with disabilities is handicapped.
Grammar aside, both handicap and handicapped have fallen out of favor in all uses relating to physical and mental disabilities. Handicapped is not as offensive as some euphemisms, but it is considered inferior to words like disabled and impaired.
Terms like handicapped parking and handicap-accessible are questionable to anyone who pays close attention to language use, but phrases like these are common in official disability-related terminology.
He was bidding for a hat-trick in a 12-furlong handicap at Newmarket last October but failed to get home in soft ground. [Mirror]
[H]er personal transparency has been a professional boon instead of a handicap. [Forbes]
Whatever may handicap the U.S. economy (such as excess government spending), manufacturing isn’t it. [Globe and Mail]
The third school was for handicapped and deaf children. [Miami Herald]
He met Lorna May Criddle, an art teacher working with handicapped children, at a dance in Norwich in the summer of 1939. [Sydney Morning Herald]
Thieves are targeting the vehicles of handicapped drivers in Metro Vancouver. [Vancouver Sun]