Vatican City is part of the Schengen area and therefore citizens of Vatican City or persons with a valid (multiple entry) Schengen visa or a residence permit for Vatican City, do not need to apply for a visa for the Netherlands, another Schengen country or the Caribbean part of the Netherlands.
In case you do need an airport transit visa (for example because of loss or theft of your residence permit for the Netherlands), you can send an e-mail to the Netherlands embassy in Rome, Italy in which you explain your situation. Someone of the embassy will contact you for further instructions.
Your application will then be processed by the Luxembourg embassy in Rome, Italy (information in Italian).
Please note that a Schengen visa will not be issued for tourist purposes in case the residence permit for Vatican City is expired or in renewal.
In case you have a question, you can send an email via the contact form.