There are certainly hundreds of shops available. However, I only list those which I used before successfully.
When I am looking for new sources I watch out for:

• Price
• Customer Service
• Quality

In my experience there is no shop that covers everything. There are shops that can provide everything but that does not mean that the quality is always the best you can get (in the same price range!).

Shops

• Asahi Archery
  The website is not very appealing (it is improving though) but the customer service is ace. Just contact them by This e-mail address is being protected from spambots. You need JavaScript enabled to view it in english and tell them what you are looking for. Usually you will get an answer within 3 days with all the information you need.They take all credit cards and usually send the order via EMS.

• Bu Jin Design
  Here you get very good hakama in heavy textile fabric for a reasonable price. The advantage of such hakama is its durability and aesthetics.

• Thoralf Schuchardt
  This site is in german!
  You will not find bamboo equipment there but they have good premade aluminium arrows and arrow parts (shafts, tips, notches, feathers...)
  

• Sambu Kyuguten
  They have a very nice website and some extra stuff which Asahi Archery does not list. Long delivery times!!!

• Dick
  You will find all kind of high quality tools there along with working materials that are hard to come by - like urushi lacquer etc.

General Information

• Customs
  Be aware that the prices listed on the websites are always with customs. Depending on your country you have to add a certain percentage of the complete value of the package (that includes shipping costs!). For Austria customs is approx. 25%.

• Shipping Methods
  They usually offer either EMS or Air Mail for oversea shipping. Please be aware that only EMS has an insurance against loss and damage. Besides, it is a lot faster too.

• Buying high quality stuff outside Japan
  That is not recommandable unless the shop has a branch office there. I can guarantee you that it is cheaper to buy directly from Japan. And usually the quality is higher as well.

• Be very specific
  Always be as specific as possible. Rather keep a communication going until everything is clear than getting the wrong stuff. If you feel uncertain asked more experienced kyudokas for help. Especially when you buy bows and arrows. Usually you can exchange ordered stuff but you have to cover the shipping costs yourself and they hurt for bulky material.

• Be polite
  Especially when ordering from Japan be polite. You can asked any question and you will get an answer. But courtesy is part of the japanese way of life!