Can corresponding authors include their personal website to be contacted?
Instead of including a personal website, why not just include a permanent e-mail address? 


Simplest would be just get a Gmail account (or any free webmail provider) for your work correspondence. 
If you want a more professional looking account, sometimes professional organizations can provide to dues-paying members e-mail forwarding services. For example, the American Mathematical Society offers precisely one such service. 
Also, if you have a personal website, maybe you can set up a mailbox under its domain that you use to receive work correspondence?


One other thing you can do is to sign-up for a digital researcher identification so that your publications are properly linked to each other. That way readers may be able to track down your more recent publications and find your contact info there. 



But to the question you actually asked: I am sure it depends greatly on journal and publisher, so you are much better off asking them than us.