Amazon is a good place to buy adult diapers and they ship out fast.
I can order a case of adult diapers on Amazon and then they will sometimes arrive at my door the next day or at least within a few days.
I buy a lot of stuff through Amazon and they are great with shipping etc and usually deliver the diapers in an Amazon box.
Although just note that if you do buy adult diapers through Amazon that sometimes they may be sold by a third party Amazon seller and they may not ship discretely which has happened to me before.
It's not a problem for me since I live alone but if you live with your parents or want to make sure the diapers are delivered discretely to your door then you need to make sure they will ship discretely.
Abuniverse.com bambinodiapers.com and healthykin.com are all good places to buy adult diapers as well and even ebay.