If your website has a contact form, it needs a Privacy Policy.
Contact forms ask for a “name” and “email”, which are examples of “Personally Identifiable Information” (PII). Multiple countries and states have enacted privacy laws that impose heavy fines for not having an up to date compliant Privacy Policy. Also, over a dozen states are proposing laws that can apply to businesses regardless of their location. Several of these proposed laws will enable its citizens to sue businesses of any size located anywhere.
It’s simple: if you ask for PII through a contact form (and you want to avoid fines and lawsuits) then you must provide a compliant Privacy Policy.