No, I couldn't find his grave in any National Cemetery or even in Find A Grave. The entire National Cemetery organization does not show a record for him under the original spelling or the newer spelling or your surname.

First, your ancestor used his original spelling while a member of Company C, 5th (Old) Cavalry Regiment Missouri State Militia, and his records call him "Private Loyd Bunjard." Yes, the online record of his service at the Missouri Secretary of State's Office does state that Loyd died about a month after he was mustered in the unit at Rock Port 13 March 1862, and that he died of measles and congestion 15 April 1862 at St. Joseph, Buchanan County, MO.

I would suspect that Loyd was buried somewhere in the St. Joseph area, but you would have to research to see if anyone there has a record of it.

Sorry I couldn't do more.

Bruce Nichols