A one-time alarm will sound, even if the Wi-Fi/Internet connection is down. However, a repeating or recurring alarm can be adversely affected by the loss of Wi-Fi connection for an extended period of time.
When one occurrence of a repeating alarm goes off, the occurrence is removed from the clock. Amazon Alexa will send a command to the clock to set up a new occurrence of the repeating alarm within 24 hours before it's due. Depending on the timing of how the operation is carried out, the operation of setting up a new alarm occurrence could be disrupted by a down Wi-Fi. Without a new alarm occurrence being set up, no alarm will go off.