Is it permissible to perform Tayammum due to the unavailability of water for Wudu' at the university, knowing that the prayer time will elapse if I wait to return home, and how should purification be performed if I am wearing shoes?

Ablution (Wudu') is a condition for the validity of prayer. If you do not find water after exerting effort in searching for it, and you fear missing the prayer time (the preferred time), then perform Tayammum (dry ablution) and pray. Tayammum is not permissible when water is available, as it is a purification necessitated by necessity.