Green’s Dictionary of Slang

(US) to get married.

1936		J. O’Hara ‘Saffercisco’ in New Yorker 11 Apr. 14/1: [S]he married Bobby Waterman, the writer. [...] They got veiled up and were together about two years.
1938		J. O’Hara Hope of Heaven 138: ‘[T]hey go and live in Ohio or someplace and veil up and live respectable lives for fifteen years’.
1940		J.H. O’Hara Pal Joey 15: It was [...] only a question of time before Joey and Jean would veil up.