The slang dictionaries I consulted identify several meanings of wally (or wallie), but none of them are directly related to groundhogs. In addition, two generally reliable mid-twentieth-century reference works for U.S. regionalisms and dialect—A Dictionary of Americanisms on Historical Principles (1951) and American Dialect Dictionary (1944) have nothing for wally, walley, or wallie. Evidently, wally for groundhog is either fairly recent or extremely localized slang.
Here is the entry for wallie/wally in Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960):
wallie [or] wally n. 1 A youth whose hair has a patent-leather luster; a ladies' man; a gigolo. [citation omitted]. c1920; obs[olete]. 2 A town sport or gambler. Hobo use. Obs[olete]. 3 A short, fat person. Some student use c1930.
And here is the corresponding entry in Barbara Kipfer & Robert Chapman, Dictionary of American Slang, fourth edition (2007):
wally or Wally or wallie n 1 Stupid person; moron 2 An awkward person; KLUTZ 3 An unfashionable person : [citation omitted] 4 The penis; DORK, PRICK : [citation omitted] ...it is attested in the US in the early 1900s meaning "A mall-town sport," and in 1915–22 in college (Bryn Mawr) and flapper slang meaning "a goof with patent-leather hair"
None of those definitions, except maybe "A short, fat person," seems transferable to a groundhog.
One possibility is that some people (perhaps at the level of individual families) picked up the usage from a series of stories by Hugh Pyle with titles like Scamper Squirrel (1973), Adventures in Animal Land (1974), More Adventures in Animal Land (1979), Mystery Stories from Animal Land (1991), and Scamper Squirrel Goes to Camp (1997), all of which include a recurring groundhog character named Wally Woodchuck.
Pyle seems to have been a one-man Wally Woodchuck industry, but he wasn't the first author to give a character that name. Olive Roberts Barton, Nancy and Nick in Scrub-Up-Land (1921) involves a quest by two children to find and clean up "Grubby Groundhog"—a quest that Grubby temporarily foils by presenting himself as "Wally Woodchuck."
Woodchucks being what they are (namely, an animal that starts with the letter w), it isn't surprising that more than one children's book writer has named individual woodchuck characters Wally. Whether that explains the term wally as your family came to know and use it, I cannot say.