This might be a long shot, but sometimes realtor.com does not pull the listing information correctly. I’ve had it happen myself in which a house with 2.5 bathrooms was listed with only 2 on realtor.com. I had to re-enter the listing on our MLS and re-save it before it would use it correctly. You could see if your agent ever showed those houses or contact their listing agents to verify since it very much affects the comps.
My experience with appraisals/assessments is in line with lenapesadie. Our tax assessor’s office has a value that is nowhere near market value and is usually updated about six months post-real estate transfer certificate. Our assessor’s office doesn’t ask or care what property has sold for; our state is a non-disclosure state so the purchase price is not public information, though any mortgages are recorded with the county. This is also why Zillow doesn’t work for all states; it doesn’t know what property sells for here unless the buyers authorize and share that information so Zillow is having to make educated guesses. Our appraisers usually mine the sold comps as recent as possible from our MLS service which lists the sale price, date, type of sale (cash, conventional, etc) and any concessions given. As with most things real estate, things are different from one state to the next.
OP it sounds like your agent has it handled. In this situation I have seen the agent appeal to the appraiser and if that delicate little situation is handled like unicorn feathers so as not to infuriate the appraiser with the slightest suggestion of inaccuracy (we are extremely short on appraisers in my area!) progress can be made. If the value doesn’t move then you and the sellers can negotiate how to meet in the middle if you are both willing. Sounds like you are close!
It IS a big deal; even when this is my job I remember that for our clients this only happens a handful of times in most people’s lives. This time from appraisal to closing can be the most stressful! I always tell people one step at a time, pace yourself!