Learn German: Wir kriegen heute neue Strähnen gefärbt. - We are getting new highlights dyed today.

This is the Recipient Passive (also called the Dativpassiv). It is used when the person receiving an action becomes the subject of the sentence. If we used werden, the sentence would be Uns werden heute neue Strähnen gefärbt, keeping the receiver in the dative case. Using kriegen allows us to make the receiver the nominative subject (wir).

What case is neue Strähnen in this sentence?

It is in the accusative case. This is a key feature of the Recipient Passive. In a standard werden passive, the original accusative object turns into the nominative subject. But with the kriegen passive, the person receiving the action becomes the nominative subject (wir), and the direct object simply stays in the accusative (neue Strähnen).

Could I use bekommen or erhalten instead of kriegen here?

Yes, grammatically they work exactly the same way. However, kriegen is very colloquial and common in spoken German, which fits the everyday context of a hair salon. Bekommen is standard and works anywhere. Erhalten is quite formal and would sound too stiff for a casual sentence about getting highlights dyed.

How would this sentence look in the active voice?