One quick warning. You do not have the same consumer protections to dispute charges on a business card that you do on a personal card.
Also, even if you see the signup bonus offer, they can DQ you from it if you already have the "wrong" kind of Amex. At one time Amex was incredibly liberal with signup bonuses, they have done a 180-degree flip.
I've never had a problem getting a biz card. I got one for my father, whose business consists of occasionally driving people to the airport. And got one for my wife, who has no business at all. It makes it look a tiny bit better if you have an Employer ID number, but you can get that online from the IRS in 5 minutes, and no, it won't cause them to come around and say "Hey, how come you aren't paying taxes on this business of yours."