Amazon.com Inc. has made it easy for small brands to sell their products to large numbers of customers, but that has also enabled some counterfeiters to cut into their business.

Sassa Akervall gets much of the sales for the SISU-brand mouthguards that her family invented from Amazon. The Michigan-based entrepreneur said fake versions of the product on the site have undercut her price and hurt her business. She has reported the problem repeatedly to Amazon, but the fakes keep resurfacing.

“It’s...