The company and order books of Captain Arthur P. Hayne’s Company, 1st Regiment of US Dragoons, may exists in the Records of United States Army Commands 1784-1821, Record Group 98, at the National Archives. You will probably have to visit these records yourself or you will probably have to hire a researcher. I do not believe that the NARA does this type of research. I hope that these records are indexed.
Another good possibility is if Captain Hayne donated these books after the war to the South Carolina Historical Society, the state library or a local historical society. Many of these societies have indexes on-line. Writing to these societies would be the second step.