I would like to purchase Lion, but the Mac App Store requires a credit card, and I don't have one. Are there any other options which don't require a credit card?
Just to note, you can (at least in the UK) use the App Store with a debit card.
However, if you don't have one of those either, and you're not having any issues with OSX 10.6, why not continue using it for now? In August, Apple will start selling 10.7 on a USB drive, which you can purchase with cash in an Apple Store.
As someone already answered, you could wait till August and buy Lion in a USB Flash Drive, but that will cost you $30 more, besides the wait.
You could buy iTunes Gift Cards to put some credit in your iTunes Store account (and even open an account using using an iTunes Gift Card as the payment method). If you don't have a place nearby where to buy them, you can buy through eBay and pay with PayPal using your debit card.
Some vendor will send the iTunes Gift Card in a matter of minutes, through e-mail, and don't charge much more than in a store.
At least here in Finland they accept ClickAndBuy, which in turn accepts some debit cards (like Visa Electron). You can also add funds to the account with a bank transfer.
I used a prepaid MasterCard; you go to your local bank & you buy one, they come in denominations of €50,75,100 (or your countries equivalent). They work similarly to cellphone top-up card (for prepaid contract obviously)
lpacheco has the simplest solution, buy a gift card, you can get these everywhere from bookstores to supermarkets etc. Create an iTunes account if you don't already have one, and load the credit up.