There are several claims of coinage. The term appears to have become popular in 1946.
[This entry was assisted by research from Fred Shapiro and John Baker of the American Dialect Society.]
litterbug (plural litterbugs)
1. A person who tends to drop litter and not clean it up
(Oxford English Dictionary)
1947 N.Y. Herald-Tribune 16 Feb. 2/7 (heading) 47,000 subway '*litterbugs' pay $107,000 in fines in 1946 drive.
4 April 1946, Olean (NY) Times Herald, "Cleaning Up After the Park 'Litterbug'" (editorial), pg. 20, col. 1:
Warm days of late March opened a new season for the city parks, so far as attendance is concerned. They also served to center attention to a growing nuisance, the Park Litterbug. The Litterbug may be either male or female. He is deliberately or thoughtlessly untidy. He can be tracked through River Park by a trail of paper, orange peel and other debris as easily as a hound follows a fox."
21 April 1946, Ogden (UT) Standard-Examiner, pg. 2:
The season of outdoor activities is here. And that serves to center attention on the fact that litterbugs will become a part of community life just as they always have before. (...) Litterbugs are deliberately thoughless [sic] and untidy. They can be traced through city parks and along streets and roads by a trail of fruit skins, ice cream cartons, candy wrappers, milk and beverage bottles (often broken) and other debris, as easily as a hunting dog follows the trail of a game animal or a varmint.
26 February 1979, Associated Press -- Alice Rush McKeon, a conservationist and roadside beautification advocate believed to have coined the word "litterbug," is dead. She died Saturday at her home here at age 94. Mrs. McKeon first used the word litterbug to describe random discarding of trash in a nationally circulated booklet on conservation, "The Litterbug Family," in the 1930s.
5 February 1998, CNN Today -- The cartoonist who created the subway signs years ago, Amelia Opti Jones, also known as Oppy, started in the 1940s. Her son survives. She coined some words. Interview-Bill Jones, Oppy's son, says "litterbug" has become part of the language.