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Dear Ms. Forbes,
Thank you for sharing your question with the History Hub.
Several series from National Archives’ Records of the Veterans Administration [VA] (Record Group 15), such as General Index to Civil War and Later Pension Files and Case Files of Pension Applications [...], may contain information on the William Paxton you are researching. You may also be interested in the series Registers of Enlistments in the United States Army, 1798-1914, which is part of National Archives’ Records of the Adjutant General's Office (AGO), 1780s-1917 (Record Group 94).
These records are in the custody of the Textual Reference Branch at the National Archives in Washington, DC. You can contact them directly (firstname.lastname@example.org) with questions about researching and accessing records in these and related series.
As Mr. Shockley indicated, many of these records have been digitized by Ancestry or Fold3, and you can access these sites free of charge from any National Archives research facility. In order to narrow down the search and identify records related to the particular William Paxton you are interested in, you will--again as Mr. Shockley noted--most likely need to know Paxton’s full name and/or middle initial and his date and place of birth.
Additionally, the National Archives has compiled this list of Military Resources: Mexican War, 1846-1848.
Best of luck with your continued search, and thank you, again, for posting your question to the History Hub.
You can also hire an independent researcher to request and scan a copy of that file for you, and search for and copy the military enlistment records and unit details.