In general, "deny a spot" means that someone is prevented from having access to a place. Being denied a spot can be an active or passive process.
"Tony arrived at the restaurant on time, but Susan denied him a spot at the table."
"Tony arrived at the restaurant late. All of the seats were taken, and he was denied a spot at the table."
Usually, being a denied a spot refers to a location, literally. In the example you gave, it is a more figurative. His name would be prevented from appearing on the ballot.