Students admitted to a summer language institute are responsible for making their own living arrangements.
University Housing
You can find rates and application information on the Summer Term Housing website: https://www.housing.wisc.edu/residence-halls/assignments/summer/.
Off-Campus Housing
There are also off-campus housing options available throughout Madison. A good starting point for finding off-campus housing is the Campus Area Housing website: https://campusareahousing.wisc.edu/.
Additionally, Madison is home to a variety of housing cooperatives. You can find out more information about Madison cooperatives here.