State lawmakers across the country are increasing scrutiny of apparel companies, with proposals aimed at boosting accountability for the fashion industry’s labor and environmental practices.

Legislators in Massachusetts and Washington are drawing inspiration from New York legislation to require fashion sellers to map their supply chains, identify risks, and mitigate harm to workers and the environment. The New York measure filed in both chambers, dubbed the Fashion Act by supporters, has yet to move through the statehouse but has support from labor unions, environmental groups, and companies and brands such as Eileen Fisher and Patagonia.

Backers across states point to …