Is the organization a registered nonprofit? If so, I hope you're not you using a personal bank account for your finances. If you have a business bank account, ask your bank about credit card processing.
If you are not a nonprofit, do you have a fiscal sponsor that could process the credit card donations for you?
If you are a registered nonprofit, you might be able to process the donations through
Network for Good or
JustGive. There's no setup fee or monthly fee, so you can get started immediately. You might even find that you have a giving page already set up. You will pay a 3% processing fee, though. I don't know whether they require the CVC number.
Otherwise (unless someone with personal experience responds), you will have to do your own research to find an online donation processor that will work under your circumstances. Some places to look:
Affinity Resources list of Online Donation Engine Providers
Idealware's article on Selecting an Online Donation Tool
If you want to pursue the merchant account/credit card terminal idea, I suggest calling banks in your area, or calling other local nonprofits to see who they recommend.
Robert