If you are having trouble signing up for MWS through the app itself, try following these steps:
Visit https://sellercentral.amazon.com/gp/mws/registration/register.html in order to sign up for MWS*. Then select the third option, “I want to give a developer access to my Amazon seller account with MWS.”
Enter "SellerEngine" for the developer name and 4958-2614-2458 for the account number if you are in North America. European users should enter 1503-7264-5235 and Japanese users should enter 3471-0179-0453.
After you complete the rest of the steps and reach the final confirmation page, email your SellerID/MerchantID to firstname.lastname@example.org. Note that your SellerID/MerchantID isn't your seller nickname. It is a long string of letters and numbers. If you're not sure what to send, just copy and paste everything or send a screenshot of the page.
Note: These steps still require you to have a Pro Merchant account.
*If this link doesn't work and you sell on Amazon.co.uk, please visit this link instead: https://sellercentral.amazon.co.uk/gp/mws/registration/register.html