Unless you have created an account with the Vendor - that either uses a Hide My Email address, or your real Mail Address - it is very unlikely that you will be able to sign-in to the Vendors website to verify your order status.
Hide My Email addresses can be used in exactly the same way as real email addresses when creating and account with a website - or can be be similarly used just to receive email without creating an account with a third party website.
For practical purposes, the only difference is that the third-party does not have your real email address - and subsequent email can either be filtered or, if needed, completely dropped by deleting the “disposable” Find My Email address.
Additional information can be found here:
What is Hide My Email? - Apple Support