Congratulations on finding a home that you are in love with (one of you at least). As a buyer, you will certainly be more emotional about the property and the transaction than the appraiser. If the property was not offered for sale prior to your contract, I would be interested to know how the owner and listing agent arrived at their list price. What did they compare the property to? What did you compare the property to? Did you (or your agent) research the sales and/or prior offering history of the property?
I am a broker and appraiser, and sometimes find that when there is a situation like yours, where the property was not listed prior to contract, or was for sale by owner (FSBO), the seller may be a little (or a lot!) proud of their property, which is sometimes understandable... like if they had it custom built, invested a lot of money in renovations, or they need to net a certain amount to move on... none of which have to do with market value or what a "typical" buyer would pay for the property. A bank appraisal is virtually always based on fair market value (unless the appraiser/client address another scope of work - another definition of value.
Get a copy of the appraisal when it is complete. Read it thoroughly, including the "fine print". There is (or should be) a statement of limiting conditions and certification which will include the definition of market value - in which it speaks of a "typical" buyer. That is sometimes the difference between an appraiser's opinion of value and what a buyer / seller agree on, as that particular house may be perfect for you, but ho-hum to the majority of buyers.
While the bank (and seller) may consider the appraisal "gospel" once it is complete, as mentioned earlier, appraisers are human too. I've seen some good appraisal work, and some terrible appraisal work. If the appraisal is a problem, I would suggest you get a copy of the appraisal, go over it, then review it with your agent (hopefully a buyer's agent) â€“ in your case a Dual Agent or Seller Sub-Agent I presume, and then maybe contact your mortgage loan officer and ask some questions if you see anything questionable on the appraisal. Your agent should be helpful (depending upon the type of agency), as they are supposed to know the market, and the lender may / should be helpful, as they should be familiar with appraisals (at least the underwriter should). If you cite error(s) or suspect work, it may well be worth the time and money to hire another appraiser (with references... I suggest someone with years of experience in that immediate area, and/or a member of the Appraisal Institute, the American Society of Appraisers, or the National Association of Independent Fee Appraisers. Membership and / or designation shows that they are likely not "form-fillers", which is part of the reason for the mess we are in (another topic).
The short answer is... Ask questions about the appraisal process, get a copy of the appraisal, do your own homework on the market and prices of competing homes, and demand good service! A low appraisal may mean that the transaction falls through, but it may be the best thing that ever happened to you financially, especially if we are talking about a million dollar home! A low appraisal will likely mean that the price has to be reduced to the appraised value (if the seller wants to close and the buyer wonâ€™t pay the difference) and you save some money. BEWARE: Whatever the value, get a copy of the appraisal, read it, ask questions, and press for answers for anything that doesn't "add up"! Just because it appraises for the sales price (especially if it comes in right on the sales price!!!) or appraises higher does not automatically mean you are blessed and highly favored so to speak. There are those among us known as prostitutes, skippies, deal-makers, etc., as they have client relationships that are "mutually beneficial" if you know what I mean (one hand greases the other)...
All that said, there are professional, ethical / honest, educated, experienced appraisers that do a great job and can help insure that you are not getting ripped off (as can a home inspection from a professional, ethical / honest, educated, experienced home inspection relative to the condition of the property). I suggest the home inspection first by the way... then you'll know whether to have it appraised or not. The appraisal is only as good as the inspection, measurement, research made, data considered, and appraisal theory applied. IF there is a value issue, then I'd say the onus is on the seller and listing agent to defend their price.
Best wishes and Merry Christmas.