In January, Mara Kronenfeld, who runs UNRWA USA—a nonprofit that helps raise money for Palestinian refugees—was shocked by what she found online. When she Googled her organization’s name, she saw ads that looked like they came from the UN, but when she clicked on them, they led to an Israeli government website. It turned out that Israel was running a secret online ad campaign to discredit UNRWA, the organization UNRWA USA supports.

This ad campaign started around the same time Israel acc...