Shatner, who played Captain James T. Kirk on the Starship Enterprise on the programme launched in 1966, proposed the name earlier this month in response to an appeal from scientists for help in choosing the names of two newly discovered Pluto moons.
Vulcan is among 21 names in the running in an online poll organized by the California-based SETI Institute, whose team of astronomers discovered the moons.
As of Wednesday, Vulcan was the clear front runner, receiving more than 100,000 out of more than 330,000 votes cast.
Shatner, who starred along with Leonard Nimoy, who portrayed the Vulcan science officer Spock on the long-running TV sci-fi show, rallied support for Vulcan on Wednesday on his Twitter account, writing: "We are approaching 120K votes for Vulcan on PlutoRocks.com! Have you voted today?"
Until now, the two moons, each of which are about 20-30km wide, have been known as P4 and P5. The P4 moon was discovered in 2011 and P5 a year later.
Before the discoveries, astronomers had identified and named three of Pluto's moons - Charon, Nix and Hydra.
Voting for the names of the new moons ends on February 25. The selected names will be submitted to the International Astronomical Union (IAU) for consideration.
"Let's hope the IAU thinks Vulcan is a good name," Shatner said in another tweet.