Yes, you can do this, but you might want to start doing the groundwork for your e-commerce business now and officially start it once your LLC filing is finished. The point of the LLC is to protect you, and you are jumping through many hoops to get that protection over the sole proprietorship (which almost anyone can set up in an afternoon).
The problem you could run into is if for some reason you get into legal trouble that the LLC could have given you some protection, but occurred when you where using a sole proprietorship while waiting for the LLC paperwork to be finished. Your lawyer would not be too happy with you, and you would not be happy with all the extra legal bills and additional financial cost for a legal settlement.
The question you really need to ask yourself is do your REALLY NEED the legal protection the LLC offers? If so then I would advise you to wait. If not, go ahead with a simple sole proprietorship, it will save you a lot of paperwork. I would get the advice of a local lawyer before you make this decision.