Dear Mr. Bagwell,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located a series titled Records of Confederate Prisoners' Applications for Release and a series titled Records of Confederate Prisoners of War Ordered to be Released in the War Department Collection of Confederate Records (Record Group 109) that may contain information for oath applications filed at Camp Chase. These records are in the custody of the National Archives at Washington, DC - Textual Reference (RDT1). For access to these records, please contact RDT1 via email at firstname.lastname@example.org.
These records are also available on Roll 7 and Roll 8 of Microfilm Publication M598 which can be viewed online through Ancestry.com or Familysearch.org. There may be a fee for using Ancestry. Instead, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/.
We also searched the National Archives Catalog and located a series titled Carded Records Showing Military Service of Soldiers Who Fought in Confederate Organizations , 1903 - 1927 in the War Department Collection of Confederate Records (Record Group 109) and found service records for some soldiers indicating that a Oath of Allegiance to the Union had been completed but not the completed Oath Application.
For Confederate soldiers held captive past May 1865, we found a series titled Amnesty Papers, 1865 - 1867 in the Records of the Adjutant General's Office, 1762 - 1917 (Record Group 94) which contains many individuals applications for pardon and amnesty. These records only begin after President Johnson’s Amnesty Proclamation on 29 May 1865.
We hope this is helpful. Best of luck with your family research!