Joe attended UMS High School where he lettered in five sports. Joe then attended Tulane University in Louisiana, playing football from 1969 to 1971. While playing for the Green Wave Bullard played defensive back, but truly excelled on special teams. In 1970 he was named the Most Valuable Back and in '71 was named Most Valuable Player. Joe was selected to the Associated Press All-South Independent Team all three of his playing years.
He broke several records in Tulane's program, many of which he still holds today. Joe holds the school record for most punt return yards in a game, most in a season, and most in a career for Tulane. Joe also holds the record for most punt return touchdowns in a career with four, as well as the longest single punt return at 92 yards. In all three years that Bullard lettered at Tulane University, he was selected as an All-South Conference player. In 1972 Joe was drafted by the Houston Oilers.