Getting a seller permit and forming an LLC out of the state.
Technically, but the fact that you will be running this business from Illinois you would be creating nexus in that state, whether you take deductions for home office or not. Since you will be forming an LLC the taxation of your business would be done on your personal level, meaning you will still be paying all the same Illinois income tax, regardless where your business is registered.
Keep in mind that although forming and maintaining an LLC in Illinois is indeed costly, by the fact that you will be able to take many deductions related to running your business from home you might actually be saving money.
Neither Oregon nor Delaware have sales tax, however, again, since you create nexus in Illinois by running your business form there you would need to register for sales tax and collect it from your Illinois buyers, unfortunately.
To sum up, although running a non-Illinois registered business in Illinois in a clandestine manner is probably an option, I would still advise you to consider forming your LLC in Illinois for all the reasons stated above.