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Hi Roger -- thanks for posting to History Hub!
Records at the Center for Legislative Archives are made available according to the access policies set by the creator of the record. House Rule VII opens most House records when they are 30 years old and Senate Resolution 474 from the 96th Congress opens most Senate records when they are 20 years old.
For the example you provide -- EC 913, 97th Congress -- it's likely we hold the communication with the records of the Senate Committee on Commerce, Science and Transportation. I can link you to the file unit in the National Archives Catalog. It looks like we hold 18 boxes of executive communications from this committee in the 97th Congress. To follow up on accessing this EC, please email us at email@example.com.
More generally, to request an executive communication you need to know in which Congress it was submitted, the committee to which it was referred, and the EC number (ECs are usually arranged numerically) -- all of this information is available in the Congressional Record. The Congressional Record is freely available online via Congress.gov (1995 to today) and GovInfo (1873 to 2016). Also depending on the type of executive communication and the year, they can sometimes be found on the website of the federal agency that produced it.
For more background on executive communications I can recommend a Congressional Research Service report titled Messages, Petitions, Communications, and Memorials to Congress. There are also two helpful pages on Congress.gov: About Communications to the House and About Senate Executive and Other Communications. And lastly I can link you to the Library of Congress research guide on presidential communications.