Good question. I don't exactly know any ways to look up that information specifically. Honestly, I don't think there's much risk of not having customers unless you live in the wilderness of Montana or something. Obviously some places will be busier than others I suppose. But in terms of supply and demand, the more rural a place is, the fewer sitters it will have (generally speaking), while more urban places will have higher demand from more people and also more sitters, so it's a trade-off. I, myself, live pretty rural and get plenty of customers to keep me busy with very little competition. Worst case scenario is you would have to lower your prices to remain competitive for that area. You could start by looking up how many Rover sitters there are in an area you are considering moving to, and compare their rates to yours. You may find that other areas charge much HIGHER than you currently do!
I think the most important thing you can do to stay viable wherever you live is to build up your profile now. Get a lot of amazing reviews, get profile badges, make sure you have a stellar profile (ask for suggestions on here or Facebook if you want feedback), and so forth. That way, if you move to a highly competitive area, you will already stand out. (This is all assuming you can change your address and keep everything else the same about your profile, which I believe you can.)