Can New Zealand/Australian academics do consulting work during summers?
This is dependent on the university. This is allowed (generally) but you have to get into contact with the designated person(s) for your department/faculty/school regarding the appropriate procedures in taking on paid consulting work while employed at the university. 

Usually you have to have an agreement/contract set up that stipulates specific rules and regulations regarding this. Most universities would have something like this available as many academics do consulting work depending on their research expertise. 

I would get in touch with HR and/or your supervisor to find out about your options.