Dear Mr. Carnes,
Thank you for posting your request on History Hub!
Title 3, Ch. 1 of the US Code covers the process of the transmittal, venue, opening and reading of votes, but does not specify the containers.
We searched online and located a recent NY TImes article that says: "...inside two bound mahogany boxes that date from the 19th century." but doesn't list further source information. We also found a 1910 article referencing the use of mahogany boxes for carrying the votes. Additionally, there is one box presented as an artifact on the Senate history website and identified as originating from the Senate Sergeant at Arms Cabinet Shop.
Based on the information we located above, this appears to be a matter of tradition and protocol for the House and Senate. Therefore, we suggest that you contact the Sergeant at Arms of the House, the Sergeant at Arms of the Senate, the Office of the Historian and the Clerk of the House's Office of Art and Archives, and the Senate Historical Office for questions about this practice.
We hope this is helpful. Best of luck with your research!