Can you set up a trigger that automatically sends out the rental agreement after a booking is entered?
Sure can! Just create a trigger and mark it go be sent "as soon as" the booking "is created". Now that will only happen if the booking is created with an email address of course, so if it's a channel booking, you may need to bring in the guest's email address before the trigger can be sent. Channel Bridge can download guest email addresses (along with a lot of other things) for you after the VRBO or Airbnb booking is imported.