NGO

OK, so where's the rain forest in Ubud?  :<)  Just kidding.

You ask,

“Can an NGO which is registered outside of Bali, legally work in Bali?”

It's my understanding that under Law No. 28, the so called “Foundations Law” and the updated regulations, “GR 63/2008” that as a legal foundation (NGO/Yayasan) and registered with The Ministry of Law and Human Rights, the non profit beneficial activities of said NGO or Yayasan can be undertaken anywhere in Indonesia. 

You also ask,

“I have been told that anyone volunteering in Indonesia must have a full Working Kitas
Is this any working kitas or only one sponsored by an NGO?”

You neither “need” a KITAS visa or a work permit to do unpaid volunteering work at any registered and licensed NGO or Yayasan in Indonesia.  I say need with quotes because the visa regulations do not specifically allow for this, rather it's more of an unofficial approach and in recognition of the fine work that many NGOs and Yayasans do in Indonesia. 

There are a large number of foreigners who regularly come to Indonesia on VOAs and 60 day tourist visas to conduct their volunteer work on a regular basis. 

Paid foreign workers at licensed NGO's and Yayasans are required to be issued work permits.

This is exactly opposite to what immigration say
Although many foreigners do volunteer for years without problems
the actual law forbids this
Maximum penalties for unpaid volunteering are quite high

Glad you caught the Ubud jungle bit