Beds can be hard to find in midsummer, even with high-season rates. The tourist office offers an accommodation booking service, which costs £4.
If you're driving, look for one of the B&Bs clustered around the outside of the medieval city walls; many include free permits for on-street parking. Within the city walls there is an excellent array of sleeping options, from stylish hostels to pub rooms and heritage hotels.
It's worth looking at serviced apartments if you're planning to stay more than one or two nights. In York Holidays offers a good selection of places from about £290 for three nights (which is the minimum) or £450 a week for two people. The Lawrance is also an excellent option.