The ban was approved in January and technically took effect in March.
It applies to grocery stores, convenience stores, liquor stores, drug stores, pharmacies and mini-marts.
Shoppers who don't bring their own reusable bag will be able to buy paper bags for 10 cents each.
The ban was adopted to help reduce the number of single-use plastic bags being thrown out.
Earlier this year, a study by scientists at the Scripps Institute of Oceanography at UC San Diego found plastic in nearly one in 10 small fish collected in the Pacific Ocean and estimated that fish are ingesting as much as 24,000 tons of plastic annually.
The study raised concerns that plastic pollution may be climbing the food chain into seafood consumed by humans.
City News Service contributed to this report.