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We responded to the email request you submitted to the Center for Legislative Archives, but also wanted to share the information on the History Hub as well.
We hold the official records of congressional committees and the records of the U.S. House of Representatives and the U.S. Senate as a whole. We do not hold the personal papers of members of Congress. So while we may not have letters sent specifically to Louis C. Rabaut, Charles G. Oakman and Barrett O'Hara, we will have letters and petitions submitted by members of the general public to Congress and referred to committee.
The addition of the phrase "under God" to the Pledge of Allegiance was introduced through H.J. Res. 243, 83rd Congress (68 Stat. 249 / PL 83-396) -- a joint resolution to amend the pledge of allegiance to the flag of the United States of America.
H.J. Res. 243, and the identical S.J. Res. 126, were referred to the Committee on the Judiciary of both the House and the Senate. There are relevant records in both the petitions and memorials referred to the House committee and the petitions and memorials referred to the Senate committee in the 83rd Congress, as well as in the legislative bill file for S.J. Res. 126 and the papers accompanying H.J. Res. 243.
I'd also recommend looking into the personal paper collections of the members of Congress you identified. The website bioguide.congress.gov is the best source for identifying where these archival collections are located. In addition to a brief biography, each entry contains a list of repositories where a particular member's papers are housed.