Unfortunately uni accommodation is very expensive (upwards of £55 per night) and limited to single rooms. There are plenty of B&Bs in York where you can find much cheaper prices, so I’d suggest you try to book there first.
Bed and Breakfasts
There are plenty of B&Bs and hotels in York that you can browse online. You can use this search tool to find accommodation near the university:
Here are some quick links of places near the uni: