The reason reach out is used is to give the individual being asked some autonomy / responsibility in the action.
Eg if you ask someone to call bobs surf shop and they don’t answer they might stop there or call back later. If you ask them to reach out they might go down there and see them, set up a meeting, speak to a friend that works there, etc.
Both Wiktionary and dictionary.com list definitions of "contact" as a verb. I don't have access to the OED at the moment, but I assume they would have it listed as a valid verb as well.
Eg if you ask someone to call bobs surf shop and they don’t answer they might stop there or call back later. If you ask them to reach out they might go down there and see them, set up a meeting, speak to a friend that works there, etc.