Yes, you can do basic customization. In this case, you could use either a custom link or a custom field. As a custom link, simply go to Setup > Customize > Contacts > Buttons and Links (Classic) or Setup > Object Manager > Contact > Buttons and Links (Lightning), click "New" to create a new link, and enter in the URL, which would look like:
Afterwards, go to the Page Layout screen, add this link to your page layout, and save the changes.
As a custom field, you'd go to the same Contact area from above, go to Fields (Classic) or Fields and Relationships (Lightning), create a new Formula field with a Text return type, and use a formula like:
'View on E-Commerce Site')
You can move this field around on the page layout as you like. It also has the advantage of being usable in reports.