They most likely DO exist, since the reflected worlds seem to be similar to the primary one - the question is whether or not they are still operational given the state of aether in a void-consumed world. In all likelihood, if we did go to the Thirteenth Reflection, they'd come up with some handwave that would allow us to use them; probably some gadget created by Cid that powers the aetherytes with the necessary juice transmitted from our world. Marketboards, though, probably not. Not that that's all that different from what we have already, with MB access only available in a few select places.
Possibly the most interesting thing about an idea like this is that the Ascians would be totally on board with it! Losing the Thirteenth Reflection was a major gaffe on their part, and if we found some way to salvage it they'd be all for it. It'd be interesting having them as allies, for a change.