Delaware vs. Illinois LLC for an Online Art Gallery
As far as I understand your business has two parts - the online sales and the off-line art shows. Theoretically, for an online only business you would be able to have a Delaware LLC, however even in that case if you maintain physical operations in Illinois you would be required sooner or later to register that LLC in that state as a foreign entity.
Given the fact that you would also be running art shows in Illinois then I would say registering in Illinois would be inevitable anyway.
As far as sales tax goes you would need to collect it on all sales to Illinois based customers, since your business is located there (just FYI Delaware has no sales tax so you don't need to worry about it either way).