Below is a record of the song found on eBay. Even better is that I found a recording of the song online here at BaseballRecordings.com. Go and check it out. Heck, sing along.
UPDATE: Here is more information about Gladys. She joined the Dodgers at Ebbets Field in 1942 and stayed until the team moved west. Originally, she was discovered by Larry McPhail as she played the organ at Madison Square Garden for New York Rangers hockey games. She would later on play at Knicks games. In addition to playing the organ she often sang the National Anthem at Dodger games. When opposing teams came out for batting practice she would often play the following tunes, per a commentator at Baseball-fever.com.
Giants - "We're Calling All Fans, All You Giant Ball Fans"
Phillies - "Pennsylvania Polka"
Pirates - "Pennsylvania Polka"
Cardinals - "St. Louis Blues"
Cubs - "Chicago"
Reds - "Down By the O-hi-o"
UPDATE II: Unfortunately, the BaseballRecordings link is now dead. Have no fear, though. Youtube has the song available for you to enjoy. Watch it at the very bottom of this post, or go to youtube here.