Can't answer for your state but just this year, California has added a new question to their tax form: (paraphrased) "Did you purchase anything out of state over the internet during this tax period and had delivered into California? If so, what is the value........"
Yeah, we have had that for a couple of years. Amazingly, nobody seems to buy anything over the Internet. I would bet adding that to the form cost more than they have collected from people who actually put something in that box.
State sales tax and the Internet is certainly a gray area. I have seen several attempts to force other states to collect sales tax on Internet sales, but the problem always comes down to enforcement, WV can't force someone in PA to collect sales tax.
As for the OP, in most cases, you can file with the state/county where the purchase was made and get the sales tax refunded. Then, as others have stated, you are supposed to file in your own state. My guess is also that Ohio will report the purchase to your home state so they will know.
I first moved to WV in a Jan or something. I filed my federal return with a WV address. The feds reported this to WV, who then filed a WV return for me on my behalf. Took me months of letter writing to get them to understand that during that time I did not live in WV. They kept telling me to get them my return from where I lived previously. Well, TX doesn't have income tax. This just confused them more. Finally, I had to get a paper from the DMV showing what date I surrendered my TX license before they would believe that I did not live in WV. Even a W2 from Dec 31 showing a Texas address wouldn't do it.
Point is, these guys all talk to each other. Keep that in mind if you try to get a refund and don't file in your state.