Not advising you to do this, but many people apply for and receive business credit cards by declaring themselves as a sole proprietor and using their SSN as their tax ID. If asked for documentation, they'll respond that they're starting up an ebay business. The bank generally is mainly interested in your credit score, i.e. your ability to pay. If you haven't had credit for long (< 5 years) or don't have an excellent credit score, it won't be worth the hard pull on your credit. If you decide to proceed anyway and get declined, immediately call the reconsideration department and be prepared to plead your case.
Note: One difference between personal and business cards is the former offers certain consumer protections not provided to business card holders. I've never had a problem with this (two business credit cards for an actual LLC), but it's something you should be aware of.