You can use a debit card online exactly as you would use a credit card. Using either debit or credit cards in an online setting usually just involves submitting the important credentials: the full number, the name on the card, the security code and the expiration date.
The reason online retailers do not distinguish between debit cards and credit cards is because they don't need to: it doesn't matter whether the payment comes from a credit line or a bank account, as long as the product or service is paid for. Many retail locations swipe both debit and credit cards the same way, while others will request a PIN for debit if the buyer explicitly states that they are using debit or request cash back. Cash back, moreover, is impossible to use in an online setting; as such, websites don't have to account for the debit-exclusive option when implementing payment options.
Of course, using either debit or credit cards online always poses some level of risk, as it does in a physical retail setting. Most established companies will only take card information within very secure and encrypted servers, but that does not mean buyers are completely safe. Consumers should always proceed with caution when giving card information, whether it be online or offline, credit or debit.