There is a lot of information swirling back and forth on this thread, figured I'd offer my 2 cents since I have a pretty solid understanding and study of credit cards at this point:
1. You do not need ANY official business information to get a business credit card. Simply your social security number and 'a business' will get you a 'business card' from Chase for instance. I know because we just got another business card for my wife last week, who simply used her social. She does not have any LLC or official business, simply has a 'side hustle.'
2. A business credit card will be a credit hit on your credit report as a hard inquiry, but will not be reported on your personal credit score moving forward.
3. You CAN use a personal card for business expenses, but not vice versa. But, this is a rule per the guidelines of the credit card provider. That doesn't mean they will hold you to it. Just pay your credit card bill, they don't check. What do they know if your Walgreens purchase is for milk (for the office) or for printer ink. That is YOUR business, they don't really care.
4. For bookkeeping, it is simply EASIER if you use each card exclusively for business OR personal, don't mix. You are 'allowed' to spend business money on your personal card, and get reimbursed by your business. There is no legal hit for doing so. You can apply for a personal Chase card, and use it exclusively for business dealings (that would help your personal credit score for instance if you're trying to build personal credit). Although, this isn't some kind of RULE. Imagine a boss who's out and about on business, forgets his business card at the hotel, and buys a laptop on his personal card, then pays it off with his business account? That is certainly a defensible action, not out of the ordinary. Just annoying for his bookkeeper!
5. As far as piercing any corporate veil, LLC's are formed for legal liability and protection, credit cards are simply to pay for things. LLC's are your choice, not a requirement to run a business. I would say that the more important element here is where you PAY your credit card out of. If you get a business card, and pay it every month out of your personal account, then try to say later that it was a business card for another business, that would probably not end well for you. But if you pay your personal card always out of your business account, and then later say 'that card was for my business' that would be no problem! "The card has better trip insurance, offers more points for what my business buys" are all great responses. But if you comingle your funds and expenditures, it will get messy.
So get a card for your business, ANY card, I would just say try to keep that card either business, or personal, but try not to mix.
Hope that helps, would love to hear any thoughts others have.