Can you buy an engagement ring online?

Yes, and it is quite a convenient way to buy jewellery.

Additional Notes:
In short, yes this is possible. Please verify the reputation of the place you are buying the ring at. You will have to get the ring sized, or have her finger measured to make sure you get the right size. I prefer to get my jewelry from my local jeweler, because if something goes wrong, I know where to go to find them or to get it right. I also know they are not going to take my money and not give me my item or not give me a bad item and then close up shop over night. With Engagement Rings in particular, you may think a ring looks GREAT online, but when it comes in it may not be what you thought it was and may not look good on her finger like you thought it would. With a major purchase such as this, I would trust a jeweler.