Hello Mr. West,
If your family’s Bible is within the National Archives’ holdings, it was likely presented as evidence in a soldier’s pension claim of some kind. Records of this kind and from this period are accessible in the Washington, D.C. location. Please keep in mind that sometimes family lore muddles history so it is possible that we do not have it.
The first step is to contact firstname.lastname@example.org to verify that we have it in the first place. Be sure to provide any information that you think may be relevant such as the time you believe it was given and the name or names associated with it. By example, the series Family Bibles Filed with Case Files of Pension and Bounty-Land Warrant Applications Based on Revolutionary War Service, ca. 1800 - ca. 1912 (National Archives Identifier 18471430) in Record Group 15, Records of the Department of Veterans Affairs, lists the Bibles by soldier name.
If and when they locate the Bible, you are free to examine it in the research room. Simply print the email the reference staff sent you as a convenience copy and come to the main building in downtown D.C. Arrangements can be made to copy or scan the Bible if you cannot come in person but there may be associated fees. Please visit the National Archives’ website for further guidance.
Best of luck to you in your research, and thank you for using History Hub!