Short Story

Currently the only way is to use the forgot password feature on the SendGrid login page. The reset-password link is delivered to the mailbox associated with your SendGrid account (be sure to set this e-mail in advance).

Long Story

SendGrid is currently one of the few possibilities you have in Azure to send e-mails from your application. If you create your SendGrid account from Azure you can start with a free plan with 25 000 free e-mails per month.

For SendGrid accounts created in Azure there used to be an option for Changing/Resetting your password:

…which is unfortunately not available any more.

Azure Support’s reply to my question:

Passwords and other secrets associated with Windows Azure Store are managed exclusively by the provider (like Sendgrid). Please request the reset/change of the password directly with SendGrid. Support contact information is almost always found on the provider’s web site main page.

Unfortunately, if you log-in to your SendGrid account created from Azure, the option to “edit” username/password is disabled (they do not have a separate UI for changing the password and they don’t want to allow you to change the username which associates the account with Azure):

Currently the only way to change your password for your SendGrid account created from Azure is to use the “forgot password” feature from SendGrid login page:

Please note that you’ll need to enter the account username (not email address) for this form to work.

The reset-password link is delivered to the mailbox associated with your SendGrid account (be sure to set this e-mail in advance in Account Details).