Aeropostale delivers everything in plain cardboard boxes. They do say Aeropostale on the shipping label but since they sell both guy and girl clothing, no problem there. American Eagle is also good that way -- plain plastic envelopes. Hollister (And Abercrombie and Fitch) are also good. They use plain, opaque envelopes.
Amazon is great for bikinis, makeup, and just about anything in between -- plain cardboard boxes or cardboard mailers.
Victoria's Secret may or may not be a problem. They typically deliver in a plain white plastic envelope that says "Newgistics" on the outside. But if anyone asks, it's perfumes and towels for your mother, you know?
Zulily is a great source of deals on women's clothing, especially if you like tunics, leggings, and workout gear. They use an opaque blue envelope that says "Zulily" on the outside, unless you order shoes, in which case your shipment will come in a plain cardboard box.
Also learn to shop Off Fifth, Neiman Marcus Last Call, and Nordstrom Rack for deals on quality clothes and shoes. They usually ship shoes in boxes and clothes in plastic bags with their logo on it.
You are young. Buy cute stuff and enjoy it!