It is currently not possible to process ATM / DEBIT-only cards online due to some technical hurdles on the encryption side, as Corey mentioned.
In terms of using an existing bricks and mortar account to process Internet sales, you would end up paying a non-qualified rate on those sales so you will be better off getting a merchant account configured for use with MOTO (Mail Order/Telephone Order/Internet) processing.
In addition, a lot of banks will not even allow you to process Internet sales - even an occasional one - on a Retail account, so it is much better to setup a separate account for your online sales.
There are a lot of good options on the market available to provide Internet merchant accounts. The same is also true for bricks 'n' mortar (retail) accounts.
A few things that you will want to make sure you ask about with any merchant account, whether Retail (Card Present) or MOTO (Card Not Present) are:
1. Find out the Qualified, Mid-Qual and Non-Qual Rates - you'll want to keep in mind that the total % cost is not just the qualified discount rate but also mid and non-qual fees that will apply on certain transactions.
2. Find out the Per Trans Cost and any AVS Per Trans Costs - for Internet merchants, the total per trans cost is always the per trans + AVS since AVS is utilized on all trans.
3. Find out the Statement Fee, any other Monthly Fees - sometimes these are listed under different names so make sure to get an apples-to-apples understanding between various providers.
4. Find out the Annual Fee(s) if applicable - some providers assess an annual fee and others don't.
5. Find out what Monthly Minimum (if any) applies - make sure to ask about this fee. For a small merchant, this can add an additional $20-25 to your monthly cost - above and beyond the statement fee and other fees. If you can find a provider that will reduce or waive this fee completely, that can be a big help.
6. Find out if there are any Minimum Contract Lengths on any of the Terminal, Software or Gateway Service - whenever possible, look for a solution that doesn't require you to commit to a long term obligation for your point-of-sale system.
7. Find out the exact Terms & Conditions pertaining to the account's Termination/Cancellation Fee - this is a very important and often overlooked area that can hold a significant amount of financial liability for merchants if it is not addressed.
8. Do your Due Diligence - see what other merchants have to say about the processor(s) you are looking into
Hope that info is helpful - best of luck in your search for the right merchant account provider for your needs!