Yes, Sleep / Sleep Analyzer is able to record your sleep data even if your Wi-Fi router is turned off. However, please note that:

• Sleep / Sleep Analyzer must be plugged in (if you unplug it, your data will not be recorded).
• Your router must be turned back on within one hour of waking up to allow your data to sync.
• Sleep / Sleep Analyzer must be connected to your Wi-Fi network during the day to carry out any updates.
• IFTTT does not work without a Wi-Fi connection.