Vaux was admitted to the bar in 1808, and rose rapidly to prominence in his profession, although he only became judge of the court of common pleas of Philadelphia about a year before his death.

Embodying the Quaker values of morality and public service, Vaux helped found Pennsylvania's public-school system (and for fourteen years held the first presidency of the board of public schools of Philadelphia), and at one time was a member of more than fifty philanthropic societies.

He was one of the commissioners to adapt the law of Pennsylvania to the separate system of imprisonment, and also to build the Eastern State Penitentiary, and labored zealously in the cause of prison reform.

He was among those Alexis de Tocqueville dedicated his book on prison reform after his travels across the United States including Philadelphia.

He published Eulogium on Benjamin Ridgway Smith (Philadelphia, 1809); Memoirs of the Lives of Benjamin Lay and Ralph Sandiford (1815); Memoirs of the Life of Anthony Benezet (1817; with alterations, York, 1817; French translation, Paris, 1821); and Notices of the Original and Successive Efforts to improve the Discipline of the Prison at Philadelphia (1826).