There is no law that says a company must extend credit to you on the basis of emancipation. Age discrimination is illegal, but I think that only applies to between the ages of 40 and 70. For example, a rental car company can decide to refuse to rent to anyone under 25, even though such people are adults. I don't think being emancipated makes you eligible to vote in national elections, either (I could be wrong).
I would suggest opening a checking account at a bank, and getting a debit card. There is really little difference between a debit card and a credit card when used properly (that is, you pay off the entire credit balance every month).