Vest graduated from Centre College, Danville, Kentucky, in 1848 and from the law department of Transylvania University, Lexington, Kentucky, in 1853. He was admitted to the bar in 1853 and planned to move to California. However, while en route, he stopped in Pettis County, Missouri, where he defended a young African-American man accused of murder. Vest's client was acquitted but soon burned at the stake by an angry mob. Vest's own life was also threatened, but he nonetheless decided to stay in Missouri permanently, settling in Georgetown. In 1854 he married Sallie Sneed of Danville, Kentucky. They had three children, two sons and a daughter.
In 1860, after moving to Boonville, Missouri, he was elected to the Missouri House of Representatives and served as a Democratic presidential elector. As a Missouri representative he was chairman of the Committee on Federal Relations. Vest served in the House until late 1861 during which he wrote the Vest Resolutions in which he denounced coercion of the South.
When the Civil War broke out Vest was a strong "states rights" advocate during the Missouri secession crisis, and eventually sided with the Confederacy. He proposed the Secession Ordinance that was passed by the Missouri legislature in October 1861. The following year, he briefly served as judge advocate with the Army of Missouri, commanded by former Governor Sterling Price. He served in the House of Representatives of the Confederate Congress from February 1862 to January 12, 1865, when he resigned, having been appointed to fill a vacancy in the Confederate Senate.
After the war he returned to Pettis County moving to Sedalia, Missouri, and resumed his law practice. It was at this time in 1869 that Vest was asked to represent Charles Burden and Old Drum in the case that would make him famous, Burden v. Hornsby.
Vest took the case tried on September 23, 1870, in which he represented a client whose hunting dog, a foxhound named Drum (or Old Drum), had been killed by a sheep farmer, Leonidas Hornsby. The farmer (Burden's brother-in-law) had previously announced his intentions to kill any dog found on his property; the dog's owner was suing for damages in the amount of $150 ($971 2015 inflation adjusted), the maximum allowed by law.