You're in luck! Mechanisms to get credit card donations online are fully mature now. There are several companies that will process credit card payments for you for a cut of the take, though some require an application fee. A few actually specialize in nonprofits. All promise security for your donors. Network for Good has a page of guidelines for evaluating online fundraising services.
Here is an example of a page that was built for absolutely no money, using one of the fundraising options available to charities and non-profits. Incidentallly, it's a worthy cause. You might want to chip in a few dollars to the Missionary Volunteer for the Deaf
The cheapest and simplest way to receive donations is to use a service such as Amazon's Simple Pay Donations that collects the money on their site. They give you a "click to give" button to put on your website, which takes a donor to Amazon's website, where he uses Amazon's regular payment forms to make his donation. The visitor can use whatever credit card he usually uses to buy stuff on Amazon. American Express has a version for Amex card owners. For Amex, your charity need to be listed in Guidestar's database as an eligible recipient.
Network for Good is an even better choice than Amazon if your are a U.S. 501c3 with a tax id number. Although there is no database to download, they do send you a copy of the email receipt they send to each donor, so you get the info, but you will have to reinput that information into your own database. The advantage of using them is that there's no setup fee and it's easy to get started. They're a great choice if you don't anticipate getting too many online donations. For an example of a site that uses Network for Good, see The Ossining Food Pantry.
There are companies, such as Donate.net that will host a customizable donation form or forms on their site, process credit card transactions and allow you to view and download all the information about your donors. These services allow you to have a professional looking donation form just like the big guys, and to guard your own donor information.
Although there is some effort and expense involved in setting up your form (Donate.net charges $200 for setup), it's worth the extra expense for all but the smallest organizations.
This is a real win/win. A shopper signs up at an affinity site such as iGive.com, frequently getting coupons or discounts as a signing bonus. When he buys something online, part of the purchase price goes to the organization he designates. For the organization, this is free money. The only effort on the organization's part is asking their supporters, both online and in the real world, to patronize the affinity site's merchants. Another such site is Goodsearch.com. If you can get your supporters to use their search engine and their shopping site, they'll chip in part of their profits to your organization.
There are search engines, such as Kanoodle that charge people to advertise. They in turn seek affiliates to display their search boxes, so that their advertisers can get "clicks", i.e., people clicking on the sites that are listed when the visitor to the affiliate's site enters a search term in the affiliate's search box. The only expense involved is the few minutes it takes to sign up and add the search box to your pages. Here's a good idea: The Rainforest Site has a tagline above their search box that says "Each search preserves over 10 sq. ft."
CafePress allows you to set up a store selling your own logo'd merchandise at absolutely no cost. Not only will you make a bit of money on each item sold, the people wearing your items will be giving you free advertising. CafePress takes care of all manufacturing, order tracking and shipping. All you have to do is design the items and collect the money.
Reader's Digest lets you sell magazines online through their www.eFundraising.com site. You can ask your supporters to help you by purchasing or extending their subscriptions through you. No cost to sign up.
Amazon has a particularly nice ad program. (See this page for some examples.) You can select the books or other merchandise you offer so that it's relevant to your site, or just have a general link to Amazon. Of course, you would want to place some text above the ad asking visitors to support your cause by using the link to purchase merchandise. Google has nonintrusive ads -- like the ones at the bottom of this page. Theirs is a variation on the Pay Per Click paradigm, so you get a few cents each time someone clicks on their ad. They are very good about targeting ads to your website content. So, if you are an environmental organization, they might have ads from the Discovery Store or Cabela's. Unfortunately, they may also have ads for rival organizations.
You can actually sell ad space on your site. AdBrite functions as a buyer/seller marketplace, letting you set your own terms. Several nonprofit sites have a "Visit our sponsors" area, with a request that you help the cause by patronizing them.
For more information on fundraising on the internet, try these sites:
The Nonprofit Matrix
Articles about online fundraising, guide to affinity shopping, portal sites, etc.
Techsoup Articles and online community for nonprofits.
Internet Nonprofit Center Offers some articles and a list of resources