This is a question of brand building so the question you need to ask it what am I promoting?
If your main objective is to push the website then you must have the website logo on there. If the main objective is the business then the business logo but if the objective is "well, to be honest I want to push both" then you need to think about having one logo for both or to re-ask the question.
While there is nothing wrong with having CompanyX the power behind WebSiteY on your card one is trading on the brand capital of the other. And therein sits your problem.The more brands you want to build the more work you have to do.
One solution is to have a number of cards. For example I often carry cards that say "Lord Matt - SEO and Geek", Cards that say "Matthew Brown, Public Speaker, Author and Tech Guy" and ones that say "Thanet Star: Thanet's leading news site".
I have been known to give one, two or all three to people depending on context. When I am business networking in Thanet I tend to opt for the Thanet Star cards (especially if I am researching a story) but if I think the person might be more interested in me as an author or public speaker I will give them the one that has a Photograph of me on it - after all I want them to recall who I am and what I do.
If you are willing to do the leg work to build both brands than the brand need a card each. Otherwise ask yourself which is the more important brand right now (my guess would be website by every case is different).