York is a good-sized city with about 200,000 people, but nothing like London or even Leeds (nearly 800,000). While you wouldn't want a car to get around within central York, much of which is pedestrian streets anyway, you wouldn't have much trouble driving in and out of the city to the highways and roads around it. The key is a hotel or B&B that provides parking (preferably free or at least low cost), or is close to a public lot where you can park overnight. For example, the Abbey Guest House has limited free guest parking but is right next to a big public lot that was crowded in the daytime but had plenty of overnight space as I recall. Most accommodations that don't provide parking will be reasonably close to parking and their websites will explain where.
Since York isn't such a big city I'd be surprised if there were any really convenient public transportation to and from outlying areas or towns. Of course there is good bus service, used by many commuters I'm sure, but you should be able to find a place to stay that lets you park nearby and also walk to the main sights.
GPS is always helpful driving in Europe. Rental companies provide it for a charge that adds up fast on a daily basis, or you can buy a unit for under $200 at home, load it with a Europe map (under $100), and use it for years on both continents. It's especially helpful navigating through cities and towns. Best used along with a good map and a navigator (you or your sister). I'd recommend GPS and a map whether you stay in York or elsewhere.