Does an Amazon Prime membership qualify as a business membership fee?

Yes, IF you used it exclusively and regularly for your business.

Hi,

You've used this response in several places here.

Please cite your source. I'd like to see the full text.

Exclusively and regularly seems clear, but it's not.

It is possible, for example if you ordered supplies and needed the free shipping, however if you also used the free shipping for your own personal use, or you also used other feature of the Prime membership, you would need to pro-rate it. For example, 40% business use and 60% personal. Allocate the cost the same way. 

So does that mean that if you occasionally bought stuff that wasnt for your business, then you couldn’t write it off?

Your write off  will need to be prorated between the business & personal usage.