There is much more info neede before proceeding... like how much lower was the appraisal, what kind of property do you have, what bank, etc. With this info, I have been able to make my sellers thousands of dollars in their favor. Hopefully, you are working with a full time, professional Realtor who is experienced enough to safeguard your side of the transaction.
Before asking for a second appraisal, I would ask the buyer for a copy of the appraisal that they received. You can then have your real estate agent look to see if these are the most comparable sales in your area. If they are not, there may be an opportunity to contest the appraisal. Otherwise, it is solely up to you to decide whether or not to put it back on the market and see if the next buyer's appraisal comes in higher.
Otherwise, you may unfortunately need to either let this buyer go or renegotiate for the lower price.
As the Seller, there isn't much you can do, other than lowering the Selling price.
Asking for another Appraisal is really not an option, the Lender can refuse to recognize it.
The Buyer is the one who can come up with the difference or find another Lender;
Unfortunately, the Buyer is probably glad that it came in low; they have leverage to get it cheaper.
What did your CMA, by your Realtor, come in at?
Did it agree with the Appraisal?
Good luck and may God bless