One authoritative source gives the following as etymology for the various renditions of your phrase, which parallels kira000's offering:
Etymology: Possibly from "he couldn't punch his way out of a paper bag."
(idiomatic, formulaic) A minimal level of competence or effectiveness, as used in phrases where one is unable to perform such.
"My boss is so clueless, he couldn't schedule his own way out of a paper bag".
"Junior varsity couldn't play their way out of a paper bag" in comparison.
Quotations: 1923, Asia, volume 23, American Asiatic Association, page 281
"The old chap did not look as though he could push his way out of a paper bag".
1924, Harry Charles Witwer, Love and Learn G. P. Putnam's sons, page 260
"Personally I don�t think this egg could punch his way out of a paper bag. He�s only been workin� in this trap two days"