The Lambertville ECD site is pointing to a copy of Samuel, Ann, and Peter Thompson's Twenty Four Country Dances for the Year 1782, on the Vaughan Williams Memorial Library website. There are a couple of things that the term "allemande" could mean in 1782, but the one that seems correct for that version of Away To The Camp would be danced as follows:
The couple link right elbows and then straighten out their right arms to reach their partner’s hand, while they hold left hands behind their backs. (For an allemand reverse they would reverse this and start by linking their left elbows.) They dance forward once around each other and return to their places. For the Thompsons' 1782 version of Away To The Camp I suggest that both Allemande and Allemande Reverse be done, that the footwork used be a skip-change step (which would have been called "chassee forward" in 1782), and that all three couples do the allemandes (although a case can certainly be made for only the active couple doing them.)
If you are interested in dance from that era, check out the blog that my wife and I have at http://www.dancehistoryalive.com/blog/ We haven't written an article on Away to the Camp yet, but we'll try to do one soon.
Jacob Bloom