Green’s Dictionary of Slang

1. (US Und.) a beggar who backs their entreaties with the playing of a patriotic tune on an accordion or portable organ.

[1888		Morpeth Herald 27 Oct. 5/3: Bouncers, Besters, Wheezers ].
1907		N.Y. Times 27 Jan. Sun. Mag. 4: Then there are ‘the wheezers,’ so-called because they played accordions or little organs which whistle and groan as they grind out some hymn or patriotic tune.

2. a cigarette.

1943

R.L. Bellem ‘Feature Snatch!’ Dan Turner - Hollywood Detective Feb. 🌐 I torched a wheezer, had an eye-opener of Vat 69 [...].

3. the penis.

1972		B. Rodgers Queens’ Vernacular.
1978		L. Kramer Faggots 321: Think schnitzel, schwantz, sewing machine, slug, spout, sword, tom-tom, wand, wang, water pistol, weener, wienie!, wheezer!, wishbone!, worm!, Ying-Yang!

4. a cigar.

2000

B. Wiprud Sleep with the Fishes 67: ‘You got another of them there wheezers, Louie?’ ‘Wheezer?’ ‘Yeah, a cigar?’.