Can I pray tahajjud without sleeping?
No. Tahajjud is a type of Qiyam al-layl that takes place after waking up from sleep at night. Every voluntary prayer after maghrib qualifies as Qiyam al-layl (standing in prayer at night), but not every Qiyam al-layl qualifies as tahajjud.
See Dar al-Ifta response to a similar question (Arabic).
Hajj Gibril Haddad