Can your credit card company randomly pull your credit report even though the account is five years old?

Yes and they do this often. If your score is lower than before, they may up your interest rate. If your score is higher or still the same, they may offer you additional cards, balance transfer offers, etc. This is part of standard credit card terms. Even if the account is closed they can do this. This is how we all get those credit card offers that say pre-approved, etc. It does not affect your credit score. It's a soft inquiry.