Can a private non-profit organisation be a lead partner?

A Lead Partner must be a public body or a body governed by public law (according to the definition of the Directive 2014/24/EU in article 2.4).
The entity may be private but it shall be assimilated to a public body according to the definition of the Directive 2014/24/UE (art. 2(4)). In case of doubt, you can ask for your NCP advise.