You don’t have to uproot your family, just go on a tourist visa and spend a few weeks travelling around NZ, knocking on doors, talking to depot managers and suchlike. You’ll see the bigger firms’ trucks on the road and that’ll give you an idea where to knock. There are several trucking firms on the Accredited Employers list - and several more who are accredited but are not listed as such.
If you can’t visit in person, use the Accredited Employers list to target known companies with emails and your CV. Search also for other larger trucking firms and email them too - they might be accredited, but you won’t know without asking. Roadranger experience would be well received, and if by some miracle you have acquired truck & trailer and/or B train experience that’ll be great too. Forklift, hazardous goods and other relevant skills will also look good…a driver in NZ is expected to do more than just driving in most instances.
NZ is easier than Australia to get a visa, but hurry - the government has proposed changes to the employer sponsored visa classes which will make it almost impossible for truck drivers to meet the new conditions. Notably the salary threshold is going from $55,000 to $78,000.