Import Eames chair from US to New Zealand  

Not to open old wounds, but...
you might try asking Alfie Hume. He has been kind with advice to me in the past. Any fine art/antiques customs expediting firm can handle arrangements, if not UPS/FedEx.
http://humemodern.com/

Be warned about acquiring objects of desire.
You seem hot to get your Eames lounge. I was, too, for years. There's a lot to know about what makes a good one, as I'm sure you are aware, but... just so you know everything... Since I got mine six years ago, my wife and teenage daughters have claimed it as their own and I haven't spent an evening's Netflix viewing in it in well over a year!
I still love them and they certainly look better in it than I ever did so I guess if I had it to do over again... ?
I'm currently in the hunt for a womb that won't break the bank. Not quite as many offerings from which to choose, though, I'm finding.

HM
Herman Miller is very strict about exporting thier pieces. No HM retailer will allow you to ship it out of the states. The reason the NZ seller won't budge on the pricing is more than likely because it's a Vitra version of the chair, not the Herman Miller version.

Exporting to NZ
As others have said, I doubt if a legit US retailer will export abroad. You will have to have them send it to a shipper who will ship it to NZ.
I have shipped furniture from US to Asia via James Cubert (safepak@sbcglobal.net). His rates were reasonable and he knows how to pack furniture since he works out of the same building as Wright Auctions in Chigaco.
Whatever shipper you use you will have a problem if the chair arrives to you with a defect or damage.

I may have misunderstood.
I took it that the OP was intending to purchase a vintage set in the US and have it packed for shipment to NZ. I understand that HM imposes strict controls on authorized retailers, but what's to stop a consumer from shipping overseas for personal use? If all applicable duties and taxes are paid, it's likely a wash, but, if not, why wouln't one?

Eames lounge to NZ
You can try a company like MyUSA as they get items shipped to them and then freight them around the world using fedex or UPS. They also have a way of using your credit card to bypass the system where American retailers will not allow you to purchase from them if you have a overseas registered credit card.
I have had a used eames chair shipped through them with no problems to Australia, it was here in 5 days after leaving America

probably shouldn't say this...
probably shouldn't say this but i'm in UK & DWR were happy for me to buy my HM 670 from them & "assist" me with shipping. It was 5 years ago though...