Where to register online business?
If your business is conducted online, and your business will not have employees, an office, or own a warehouse or any physical property in New York state, then you certainly could file your company in Delaware.
With that being said, you might still have to collect sales tax from your NY client, because the actual product is shipped from New York. The question of whether sales tax is to be collected from consumers in any given state depends on whether your company establishes what is called "nexus" in that state. I invite you to read my answer to a question related to understanding the concept of nexus: http://www.incorporationanswers.org/thread/250/Form-a-LLC-for-online-business-using-Amazon-FBA.html.
With that being said, you can obtain sales tax ID in NY without necessarily having to register your Delaware company in New York as a foreign entity. I would recommend you to consult an accountant to get a more definite answer, but use all I mentioned above as food for thought.