You do not have to travel to New Zealand to lodge your Investor Visa application. Only after the visa has been issued will you need to travel to New Zealand.
Do I have to be present in New Zealand before getting the residency by investment application approved?
Do I need to travel to New Zealand in order to submit my application for its residency-by-investment program? Do I need to be present in the country while waiting for the processing result?
The key requirement for investor visas for New Zealand is that once your application is approved in principle, you will have 12 months to transfer your investment funds in an acceptable investment in New Zealand. To make an investment in New Zealand, you can apply for a work visa so you can travel to New Zealand to look into investment opportunities.
You can lodge your resident application off-shore and do not need to stay in New Zealand while your application is being processed. However, once your resident visa is approved, it will carry conditions that will include a minimum period of time in New Zealand required for the principal applicant.
You do not need to be present until your application has been approved in principle. Even then you could arrange for the transfer of investment from overseas. Therefore the only time you would need to come to New Zealand would be after issue of the conditional resident visa therefore to start the resident visa.