Joseph Smith’s journal mentions his nomination of James Arlington Bennet as his Vice President.

Date: Mar 4, 1844
Type: Personal Journal / Diary
Source: Willard Richards
LDS
Hearsay: Direct
Scribed Summary
Reference: Joseph Smith Journal, 4 March 1844, December 1842–June 1844; Book 4, 1 March–22 June 1844, p. 4, The Joseph Smith Paper
Scribe/Publisher: Willard Richards
People: Joseph Smith, Jr., James Arlington Bennet, Willard Richards
Audience: N/A
Transcription: previous to the following meeting— Prest. Smith nominated. <Ja[me]s> Arlington Bennet.— vice Prest.
Source Link: https://www.josephsmithpapers.org/paper-summary/journal-december-1842-june-1844-book-4-1-march-22-june-1844/6
Citations in Mormonr Qnas: Joseph Smith's Presidential Campaign