I do essentially ALL of my shopping online, so I’m going to start compiling a list of websites that ship to Puerto Rico. Unfortunately, most websites that I’ve happened upon do not, but these are the websites that I have so far:
LinensnThings.com - Bedding and home items.
Faucet.com - A great store if you’re building a house. They ship to PR, but not to P.O. boxes.
Ulta.com - A beauty supply store.
Zappos.com - The webs biggest shoe store (plus sunglasses, clothing and more).
PetEdge.com - A huge selection of pet supplies.
Target.com - Target ships some items to Puerto Rico. Although they say they don’t ship to a P.O. box and then sent some of our order USPS.
LaLaLingerie.com - Womens clothing and accessories.
ViciousStyle.com - Belts, belt buckles, clothing and accessories.
WilsonsLeather.com - Leather clothing, purses and accessories.
Delias.com - Womens/teen clothing, shoes and accessories.
Cooking.com - Just about everything you could need for the kitchen.
Organize.com - Organizing solutions for the home and office.
TheSpiceHouse.com - Merchants of exquisite spices, herbs and seasonings.
SolutionAccessories.com - Changing towels and terrycloth seat covers.
OfficeMax.com - Not only does Office Max deliver to Puerto Rico, orders over $50 are delivered for free.
I will be adding more websites that ship to Puerto Rico, as I happen upon them. 
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