Yes you can, but it's not the credit card in the traditional form. Some banks (QNB, Mashreq for example) have an option where you can get a 'Fixed Deposit' credit card.

I'm sure different banks have different stipulations, but QNB do a credit card based on a fixed deposit of 10,000 of which the card limit would be 66% (approx). Therefore you have a 6666QR limit and no more. The bank is safe and you have a credit card for booking flights etc Would work on larger amounts I'm sure, but you need the fixed deposit, and I'd assume a relationship with your bank.

Hope that helps. Oh, and I didn't need a company letter, but then there's no risk to the bank.