NEW DELHI—India is tightening restrictions on foreign e-commerce companies operating in the country, in a new challenge to Amazon.com Inc. and Walmart Inc. as they bet billions on the nascent market.

Current rules forbid non-Indian online sellers from holding their own inventory and shipping it out to consumers, as is typically done in other countries. Instead, the foreign sellers have found a work-around, selling online what are effectively their own products but held by their affiliated local companies.