Maiya Jones didn’t have money saved up for a down payment when she started looking for a house in the St. Louis area last summer.

Her real-estate agent told her about a program that would cover the 3.5% required on a Federal Housing Administration-insured mortgage for those who meet certain income qualifications. She took an online course about the basics of home buying and visited an adviser. At the end of April, she closed on a $67,000 three-bedroom house and moved in with her two children.

“I...