The public waiting lists at clinics for anonymous donors can be long - up to 18 months sometimes. Occasionally clinics tell recipients that they have no donors and there too many people are waiting, so you should find your own donor. But there are donors available now if you're willing to contact them directly.
Free sites:
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http://nzspermdonors.wikifoundry.com/thread*
http://www.privatesperm.com/new-zeland-sperm-donors*
http://static3.treasures.co.nz/Community/Forum/Room/?roomId=17 (not a dedicated donor site)
Sites that require payment to contact donors:
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http://www.co-parentmatch.com/Profiles.aspx?c=New+Zealand*
http://www.prideangel.com/searchprofiles.aspx?bsc=1&profId=0&ptId=3&fName=null&continentId=0&cntryId=86&cntyId=0*
http://www.coparents.com/searchmatch.php?txtlookgender=M&txtlookcountry=NZ*
http://www.free-sperm-donations.com/donors_newzealand.htm (home AI only - not clinic donors)
Note that some donors are only willing to go through a clinic, others via home AI only, and some will do either. If you're looking for a donor to go through a clinic, there is no need for the donor to be local as clinics can transport sperm between them.
I've heard that some people on these sites are dodgy so you do need to be careful, but there are genuine donors available.