Internships / Practicum
The biggest problem with finding an internship on your own over here is that the term internship or practicum does not exist. Students do not complete anything comparable to an internship during their studies. This is why companies in general do neither offer any internships nor can they do anything with applications received for that type of work experience. 
For those reasons I chose to apply for my own internship through an organization called World of Xchange that basically acts as a type of head-hunter looking for work in the name of the intern. 
Unfortunately those internships are not paid. But let me tell you - there is no such thing as a paid internship in New Zealand. If you want to earn money you will have to go with the work experience program which is unlikely to offer you work in the field of your studies.  

Work Experience
The Work & Travel New Zealand program allows young people from eligible countries to explore life in New Zealand on a Working Holiday Visa by working in seasonal or casual jobs while traveling across the continent. I can speak for most bigger cities when I am saying there are ALWAYS jobs available. Especially in retail, gastronomy and the tourism industry. Walking down major streets you find heaps of stores, shops, restaurants and bars are posting open positions at their front doors. Inbetween there is usually just about as much work available on farms as there is within the cities. It should be no problem finding a job once you have arrived in NZ.       

Germany: CIEE
Other Countries: Check out your options at the NZ Embassy Webpage
 

Permanant or long term jobs
A lot of companies let head-hunters find their employees. Maybe try to contact one of those or check out the web for headhunting websites. Ads in local newspapers are another great starting point.