Yes, actually. A line in each vehicle spawn script can do it.
Find the last "PushActionShowHide" command before the Lights/Sirens/Move commands, and insert a "PushActionWait" in front of it, like so:
n.PushActionTurnTo(ACTION_APPEND,Rotate);
n.PushActionWait(ACTION_APPEND,5.0f);
n.PushActionShowHide(ACTION_APPEND,false);
The numerical value is time in seconds you want to delay the vehicle appearing and carrying out further actions.