Cannot update macbook purchase date

To my knowledge an incorrect purchase date cannot be corrected online. You need to contact Apple by phone.

First please determine if any of these phone numbers work for you: Contact Apple for support and service - Apple Support. If not, scroll down to where it says To find support options, choose your country or region.

What happens when you click the link? Did you use the “get support” option at the top of that web page?

What is stopping you from contacting Apple Support?