In simple terms, yes you can.

You would, however, be purchasing a business as an offshore investor, which has all sorts of implications. You can find a lot of information about what it means to be an offshore investor on these sites:

Companies office: http://www.companies.govt.nz
Tax Office: http://www.ird.govt.nz

I wont even try to explain the rules as firstly, I havent looked in to it much and secondly, it's something that only a professional should be advising you on.

What it could well be worth doing though, it putting everything in place to make the purchase, and by the time it all goes through, you could find you'll already have your visa anyway! Or, come to an agreement with the current owners to continue the business until you arrive. You'll be suprised how many people are willing to do that, and three months isnt long.