There were three girls named Carter on the rolls at the Mordecai school:
E. Ann Carter and Mary E. Carter were at the school for the last two sessions, in 1818; they're listed as being from Prince Edward County.
Martha Carter was at the school for four sessions, 1811-1812, also from Prince Edward County.
There are mentions of the Carter girls in the school ledger: In November 1811 money was received from "Andrew Jones of Mr. Williams for Martha Carter." Andrew Jones again brought money for Martha Carter in June 1812. Dr. Gloster was paid on behalf of Miss Carter and other girls in April 1813. Mrs. Gloster was paid for Miss Carter and other girls in September 1813 (looks like they might be boarding with the Glosters?). A Samuel Carter turns up in the ledger in April 1818, paying for "daughters and ward" (so E. Ann and Mary E. might well be sisters).
In the letters, there's a mention in February 1818 by Caroline Mordecai: "we hear of recruits every day, there are three Miss Carters coming from Prince George." (Caroline Mordecai to her sister Ellen, 14 February 1818, in the Jacob Mordecai Papers at Duke University) One of the three may have been the "ward" of Samuel Carter mentioned above, who turned out to have another name, not Carter. And note that they may have been coming from Prince George County, not Prince Edward.
That seems like a lot of information to go on, but... I'm not connecting any of these names up to a firm place, family, tombstone, or record of any kind. Carter's a common name, obviously, but still. Someone out there must know who these Carter sisters were.