“ If I own a physical store, and a customer comes it and uses the computer to purchase an item over the internet, would there be tax on this if the product is shipped to their house?”-->Yes. As long as you have a physical presence, as you said, in your state, you are required to collect sales tax if your state imposes sales tax to its residents. More recently, states have been attempting to counter this decision by requiring sales tax to be collected if an "affiliate" or "solicitor" of an online company has a physical presence in that state. Other states may be joining the list, but these laws are also being challenged. You need to contact Dept. of Revenue of your state.
“ I know there is no tax on online retail transactions, but would I have to declare tax for this if a lot of people started doing this?”-->It depends on the situation; if you have a business that sells products online, you may or may not have to collect sales tax from customers who order from your website. A merchant is not required to collect sales tax unless the merchant has a physical presence in that state. If you do, then you need to collect sales tax from your customers. If you are in a different state from your customers, you do not need to collect sales tax from that person unless you have a physical presence in that state.
“ I ask this because I was thinking that the IRS would declare my store as the point of purchase, but technically it is online.”--->As said above; it depends on the situation if you have a physical presence.The physical location of your business and your customer are in the same state, you must collect sales tax on any purchases made by that person.