The Verdict: Not eligible
Remote work as a visitor does not cover New Zealand employers or clients; for that you need a work visa. Look at the Accredited Employer Work Visa through the skilled migration quiz.
The visitor permission covers only employers and clients based outside New Zealand. Working for a New Zealand employer, or supplying goods or services to people or businesses in New Zealand, breaches your visa conditions and puts future applications at risk.
The Accredited Employer Work Visa needs an offer from an employer accredited by Immigration New Zealand, an approved job check, and two years of relevant experience or a level 4 qualification. It runs for up to five years and can lead to residence.
New Zealand has no freelance or self-employed work visa. If the New Zealand clients are optional, keeping your client list offshore lets the visitor route work.