How to get Amazon MWS Developer ID
Apparently, you must have an eligible Amazon Seller account in order to obtain a Developer Account Identifier, and those accounts start at $39.99 per month.
There are multiple discussions on the amazon sellers forum about this. This is probably the most on-topic:
https://sellercentral.amazon.com/forums/message.jspa?messageID=2958433
It appears there truly is no free nor less expensive option, and that Amazon simply does double-dip and charge both the actual seller and the developer just to use MWS at all.
Once logged in to your paid seller account... in the upper right corner of the screen, Settings > User Permissions, then under the text "Amazon MWS Developer Access Keys" press the button "Visit Developer Credentials".
If you already have a Developer ID but it is no longer working, you have to apply again here: https://sellercentral.amazon.com/mws/register/warning
Before applying, I suggest you study hard the information in these links. I think if you answer even one of the questions wrong while applying for the developer ID, you will be denied. Also, if you get denied, they don't tell you the reason http://docs.developer.amazonservices.com/en_US/dev_guide/DG_DataProtectionPolicy.html http://docs.developer.amazonservices.com/en_US/dev_guide/DG_AcceptableUsePolicy.html
In addition, FYI:
It used to be that anyone with a paid seller account could get developer credentials just by clicking some buttons. However, Amazon is now vetting developers with a more through process which could take up to 30 days.
You can find your developer ID using the "Sign up for MWS" wizard located here: https://sellercentral.amazon.com/gp/mws/registration/register.html
You will need to have a seller account on the "Professional" plan or higher. If you don't already have one, you can sign up here: http://www.amazon.com/gp/seller-account/mm-summary-page.html
Note that this will only work for vendors on the American version of Amazon - if your client is selling on one of the localized Amazon sites like Amazon.ca, then you will need a developer ID and seller account from the localized sites. In the case of Europe, a European account can authorize any other European account (e.g. a UK seller can authorize a French Amazon account.)