Typically, you can start shopping for airline credit cards when you have good to excellent credit history. There is no sense in applying for an airline credit card earlier, as you may not be approved for a Visa Signature or World Elite Mastercard cards, which offer better travel perks and benefits.
However, you can get a travel credit card when your credit score is 670 or higher. Travel cards are not tied to a particular airline but still allow to earn miles or point rewards toward future travels. For instance, the Discover it Miles card is not a part of any frequent flyer program, but it allows to earn miles on purchases. Cardmembers earn 1.5 miles per dollar spent on purchases. All first-year miles are matched at the end of the first year for new cardmembers. Earned miles can be redeemed fortravel statement credits or cash. The card has no annual fee and no foreign transaction fee.