“Scrounge” is the virtual twin of the last word I wrote about, “wangle.” Both mean roughly the same thing, emerged in Britain as World War I slang, and after a few decades got adopted in America.
The OED definition for “wangle” is “To accomplish (something) in an irregular way by scheming or contrivance; to bring about or obtain by indirect or insidious means (something not obtainable openly).” “Scrounge” is more specifically about getting; the dictionary defines it as “To seek to obtain by irregular means, as by stealth or begging; to hunt about or rummage.”
The OED cites a 1909 book, Passing English of the Victorian Era: Dictionary of Heterodox English, Slang, and Phrase, which defines “scrunging” as “(Country Boys’), stealing unripe apples and pears—probably from the noise made in masticating.” The word, with an added “o,” gained wide currency and a more general sense during the war. George Goodchild’s 1918 book Behind the Barrage provides this explanation:
In the category of “odd jobs” came “scrounging.” “Scrounging” is eloquent armyese — it covers pilfering, commandeering, “pinching,” and many other familiar terms. You may scrounge for rations, kit, pay, or leave. Signallers are experts at it, and they usually scrounge for wire. Scrounging for wire is legitimized by the War Office, and called by the gentler name “salving.”
As for U.S. adoption, here’s the Google Ngram Viewer graph for “wangle”:
And here’s the one for “scrounge”:
For whatever reason, “scrounge” reached parity later (in the 1960s as opposed to the 1940s), but ultimately became more (rather than equally) popular in the U.S. than in the U.K. Go figure!