Ask the people you live with to register their software in your (the pet's) name. You won't believe the amount of mail that will come pouring in.
Junk mail you say? Certainly not! Don't let the people you live with toss them. Just fill them out...include your birthdate, name of brothers and sisters -- whatever else you like. Don't lie, but claim all of the income of the people you live with as your own (as it should be). If asked for a signature, include a miniature paw print. [NOTE: Very few of the applications ask for species.] Many companies will reject your application...some nonsense about being too young or not having a social security number...however, the people who issue Shell Mastercards (Chemical Bank) and Cornerstone Mastercards (Mellon Bank) seem more than happy to issue credit to cats. Their telemarketers will even call your cats and try to sell things. Have trouble shutting up a telemarketer? Tell them that the person they are calling is currently using the litter pan and can't be disturbed.
(The odds are that the people you live with won't let you carry your credit cards around . Just one more example of humans thinking they are better than you.)