Fresh Air Vent to run only when system is on
The easiest way is to install the fresh air intake on the return side of your system. That way it sucks in when running and then is filtered, simple and inexpensive. If you want a self powered system again it is easy enough to put a pressure switch or vane flow switch in the duct so the system only turns on when the air handler is running. This is simple stuff.  Your hvac should know about both systems as they are common as safety’s.