A lot of VirtueMart users have a problem where they cannot add a product to cart or the cart delete/update icons do not work, often giving a redirect to the home page, or the cart is empty after registration. This is usually due to inconsistency of url in configuration. You must use either http(s)://www.domain.com/ or http(s)://domain.com/ - ie with or without the www. which are seen as a different domains in modern browsers. FireFox and other modern browser security will not transfer the session to the "new" url.
Make sure that siteurl and secureurl In VirtueMart configuration/security are both set to whichever format you choose and use .htaccess
at the Joomla root on the server to force all urls to use that format.
My view is that it is better to use www. as shoppers know that that is something to do with a web address, and where a site url is published in print, the standard now is to use www.domain.com
(ie to include the www). Additionally, forcing a www. url is good for SEO reasons.