Green’s Dictionary of Slang

cony-catcher n.2

also coney-catcher, conny-, cunny-
[cony n. (2)/cunny n. (1); play on cony-catcher n.1 ]

1. a prostitute; thus cony-catching, prostitution.

[UK]Nashe Christ’s Teares in Works IV (1883–4) 143: It is a hundred parts more hatefull than Conny-catching: it is the Nurse of sinnes, without the which, the fire of them all would be extinguished.
[UK]Dekker & Webster Northward Hoe V i: Your wife is a Tilt-boate, any man or woman may goe in her for money ; she’s a Cunny-catcher.

2. the penis.

[UK]Farmer & Henley Sl. and Its Analogues.

3. a prostitute’s customer.

[UK]Massinger Renegado IV i: All is come out, sir. We are smok’d for being cunnicatchers, my master is put in prison, his she customer is under guard.
[UK]Jonson Staple of News IV i: mirth.: I wonder they would suffer it; a foolish old fornicating father to ravish away his son’s mistress. [...] tat.: Marry, I would have the old coney-catcher cozen’d of all he has.
[UK]Bridges Burlesque Homer (3rd edn) 23: The old coney-catcher spoke.