score:5
You need a VWP-eligible passport to apply for ESTA, but you write
I am a US and NZ citizen. I normally hold both passports but both have been stolen from a car break in.
Since you don't have your New Zealand passport, you cannot apply for ESTA.
(Actually, you can apply for it with only the passport number, but if the passport has been reported stolen then the ESTA will surely be denied. If the passport has not been reported stolen, and the ESTA is granted, you still won't be able to travel to the US because you'd need the NZ passport got the actual trip, but you don't have it.)
The current US passport fee is USD 110, by the way.
Upvote:2
You can't apply for an ESTA
I am a U.S. Citizen with dual citizenship in a VWP country?
U.S. Citizens are not required to have an ESTA and are required to use their U.S. passport to travel to the U.S.
https://help.cbp.gov/app/answers/detail/a_id/1094/~/do-i-need-to-apply-for-esta%3F