You may find a list of NZ museums here and a directory of NZ libraries here.
A helpful source in determining which ports a particular vessel visited is "Shipping to New Zealand 1839-1889" popularly called the Comber Index after its compiler. The Comber Index doesn't contain passenger information or references to pictures, but it does give the departure dates, brief voyage details, and arrival ports and dates of immigrant ships to NZ.
The Central Auckland Research Centre holds a copy of the Comber Index on microfiche.
Although the idea of checking libraries, archives and museums seems very straightforward (though time-consuming), there are a couple of other important points to keep in mind.
If local collections at ports of interest don't yield an image, try Matapihi - a site that lists amongst other things, pictures contributed from a range of large national collections such as Alexander Turnbull Library in Wellington, which holds material reflecting the entire country, not just one region.
If a ship also visited Australian ports, a picture may be held in that country, if not here. Try the National Library of Australia's Picture Australia website that works in a similar way, reporting the holdings of many Australian contributing collections.
Secondly, you need to be careful to identify your vessel correctly. If you are lucky there will be enough information accompanying the image in the collection to be sure that you have the correct ship, but this is not guaranteed. If the relevant details are lacking then you need to be wary. Not only might there be two vessels afloat at the same time using the same name, but many shipping companies 'recycled' ship names, renaming new vessels with traditional company names as they took them into their fleet and disposed of the old vessel of that name.
The National Maritime Museum also holds a large collection of ship pictures. Another helpful website is The Ships List. Also try the New Zealand Maritime Index (vessel search) to identify pictures of ships that have been published in books relevant to NZ.
The Central Auckland Research Centre holds on CD-Rom an index listing the location of more than 11,000 ship pictures held in the State Library of New South Wales:
The above is not an exhaustive list of sources, but I hope it gives some ideas about where to start looking. If your search for a picture seems to hit a brick wall, don't give up. You never know what will turn up eventually and where you will stumble over the vessel you want.
For instance, if anybody is interested in the ship CITY OF AUCKLAND which brought out immigrants to New Zealand during the 1870s, there is a rather nice model of it in a glass case, sitting on display against the far back wall of the Central Auckland Research Centre, Level 2, in the Auckland Central Library, Lorne Street.