Your question is a little confusing but I'll give it a shot.
If you have a web site you can link from the web site to CafePress. If you don't have your own CafePress shop, that is fine, you don't need one. Look in the FAQ folder for the information on creating links. You could just create a link that searches the CafePress marketplace for your breed.
To earn money by selling the items you need to open an account at Commission Junction. That is an entirely separate organization. Once you open that account you apply to become an affiliate of CafePress. Approval is manual but fairly quick.
Once you have your affiliate id you can create the links on your external website. 15% of the sale price goes into your Commission Junction account. That 15% is held for a period of time while the sale "clears" the return period and processing time. After it clears it sits in your account until you earn the minimum to get a check.
Since the money is not intended for you, but for the rescue, you should open the account in the name of the rescue, using the rescue's tax id number. If they don't have one then the income will have to be credited to you. If the account is credited to you rather than them you will need to report it on your income tax, and pay the taxes. You can then donate the rest, and take whatever tax benefits are allowed.
When a visitor to your website clicks on the link they pick up a cookie with the CJ affiliate ID. The visitor can order from CafePress. CafePress collects the money, processes the order, makes the item, ships it and is responsible for the guarantee.
The effort by the affiliate is pretty much creating and maintaining the links. Everything else is a responsibility of CafePress and the shop keeper who creates the design.
You are not allowed to get affiliate credit for items bought from your own shop.
Diane Blackman
Experiment! Try things! Then if you can't figure it out - ask.