When Angel Taveras was growing up in Providence, R.I., in the 1980s, his youth baseball team was sponsored by Alan Jewelry, a local business, which played against the likes of Washington Park Woodworking. These days, sponsors include La Gran Parada restaurant, Reyes Market and other immigrant-owned businesses that have flourished in the city since the 1990s.

“Without immigrants and their entrepreneurial spirit, Providence and Rhode Island would be worse off,” said Mr. Taveras, who was mayor from 2011 until earlier this month...