Fulgham played five seasons for the Monarchs, redshirting his freshman season. Although he initially joined the team as a walk-on, he earned a scholarship by the end of fall camp during his first year. As a senior, Fulgham led Conference USA with 1,083 receiving yards and a 17.2 yards per catch and was named second team All-Conference USA. He finished his collegiate career seventh in school history with 128 receptions for 2,044 yards and 18 receiving touchdowns (sixth in school history). Following the end of his redshirt senior season, Fulgham was invited to play in the 2019 Senior Bowl, becoming one of the first Old Dominion players to participate along with linebacker Oshane Ximines. He caught one pass for seven yards in the game.

Fulgham was drafted by the Detroit Lions in the sixth round with the 184th overall pick in the 2019 NFL Draft. Fulgham signed a rookie contract with the Lions on May 14, 2019. He was waived on September 26, 2019 and re-signed to the practice squad. The Lions promoted Fulgham back up to the active roster on December 12, 2019. He made his NFL debut on December 15, 2019 against the Tampa Bay Buccaneers.

On August 9, 2020, Fulgham was waived by the Lions.

On August 10, 2020, Fulgham was claimed off waivers by the Green Bay Packers. He was waived on August 19, 2020.

Fulgham was claimed off waivers by the Philadelphia Eagles on August 20, 2020. He was waived on September 3, 2020, but was re-signed to the practice squad three days later. He was promoted to the active roster on October 3, 2020. Fulgham caught his first pass in the NFL and later caught another pass for his first career touchdown the following day, in a Sunday Night Football game against the San Francisco 49ers, on a 42-yard pass from Carson Wentz. The following week, Fulgham caught ten passes for 152 yards and a touchdown in a 38–29 loss to the Pittsburgh Steelers. He became the 11th player in Eagles history with at least ten catches and 150 or more yards in a game, and his 152 yards are the most for an Eagle in one of their first five games. The next week in a showdown with the Baltimore Ravens, Fulgham posted 6 receptions for 75 yards and a touchdown that put the Eagles in position to tie the game late in the fourth quarter with a successful two-point conversion. In Week 8 against the Dallas Cowboys, Fulgham had 6 receptions for 78 yards and a touchdown in the 23-9 win.