Google said a year ago it would stop its computers from scanning the inboxes of Gmail users for information to personalize advertisements, saying it wanted users to “remain confident that Google will keep privacy and security paramount.”

But the internet giant continues to let hundreds of outside software developers scan the inboxes of millions of Gmail users who signed up for email-based services offering shopping price comparisons, automated travel-itinerary planners or other tools. Google does little to police those developers,...