Amazon Prime members are eligible for free two-day delivery and discounts of up to 80% on generic medicines and 40% on prescribed brand-name drugs.
Customers need to provide some basic health information, such as whether they are pregnant, as well as date of birth, gender and insurance details.
Some questioned the wisdom of giving health data to a tech corporation.
Amazon Pharmacy vice-president TJ Parker said it was hoping to transform an industry that “can be inconvenient and confusing”.
“We work hard behind the scenes to handle complications seamlessly so anyone who needs a prescription can understand their options, place their order at the lowest available price and have their medication delivered quickly,” he said.
Doctors can send prescriptions directly to Amazon Pharmacy, currently available in the US only, or patients can request a transfer from their existing retailer.
Amazon, which acquired online pharmacy Pillpack for $753m (£568) in 2018, said health data would remain separate and distinct from that on its retail site and no information would be shared with advertisers without permission.
But former Amazon executive James Thomson previously told he could imagine Amazon offering gym equipment, specific groceries or other products based on the health data of a particular customer.
“When those types of things start to happen, I believe it will become much more apparent that we have a major major data problem here.