I just tried it, and it works.
Depending on your browser settings, the store icon may not show up for you.
Even if you don't use the shopping cart within the store, JREF can still get credit for anything you buy with very few exceptions.
Go to the icon, click it, and it will open the "JREF Amazon Store". In the upper left, there is a "powered by Amazon" icon. Click that, and it takes you to Amazon. com.
Even though you can't necessarily see anything difference, if you 'enter' Amazon this way, whatever you buy will get credited back to the JREF. The commission is between 6-8% for most items.
I 'managed' this for JREF for a couple years, and that's how it works. I could confirm by looking at my own purchases - I bought all my sons' college books, Christmas gifts, everything that way. As far as income to the JREF, it doesn't matter if you use their little shopping cart, or just go straight to Amazon via that link
Note - JREF can not see who buys what, but they get a list of the type of items bought and how much they made on each item.
ETA: This is regarding the Amazon link stuff, not the things you buy directly from JREF