He graduated from Bishop McCort High School. He attended the University of Buffalo where, as a member of the Class of 1964, he was the Bulls starting quarterback in 1962 and 1963. As a starter those seasons, the Bulls were 6–3 and 5–3–1, respectively. He set passing and total yardage records as the Bulls' quarterback. He also played baseball.
Stofa began his pro football career in the American Football League in 1966 for the expansion team Miami Dolphins. He played his first two seasons with the Dolphins, playing a total of eight games, starting two. With the Dolphins, he completed 31 passes in 59 attempts for 476 yards and four touchdowns. His first career touchdown pass was a 27-yard completion to Joe Auer in a 29–28 win over the Houston Oilers in the final game of the season.
Stofa was traded to the AFLexpansion teamCincinnati Bengals prior to the 1968 season. He was the first Bengal in franchise history, signing with the team in December, 1967. He has used the license plate "1ST BNGL.
He played one season (1968) with the Bengals, completing 85 passes in 177 attempts for 896 yards with five touchdowns and five interceptions as the Bengals went 3–11 in their inaugural season. Stofa shared time with fellow Bengals quarterbacks Dewey Warren and Sam Wyche.
Stofa holds the distinction of throwing the first touchdown pass in Bengals history. In the team's second game, he completed a 58-yard pass to tight end Bob Trumpy in the third quarter of a 24–10 win over the Denver Broncos for the first win in team history.
The following year, the Bengals drafted Greg Cook to become their starter. Stofa was waived during training camp, but then signed to the Bengals taxi squad. He was later claimed off waivers by the Dolphins.
Stofa returned to the Dolphins for the 1969 season and for 1970, the Dolphins' first year in the NFL after the AFL–NFL Merger. Stofa, playing behind Bob Griese and Rick Norton, played in just one game for the Dolphins in 1969, with 14 completions in 23 attempts for 146 yards.
In 1970, backing up starter Bob Griese, Stofa played in eight games, with 16 completions in 53 attempts for 240 yards and three touchdowns. It was the fifth and last season of his five-year AFL–NFL career.
He was inducted into the University of Buffalo Athletic Hall of Fame in 1973. He is retired after many years with Medical Mutual of Ohio and resides in Columbus, Ohio. He is also a former board member of the University of Buffalo Blue and White Club, which holds annual fund-raising campaigns to provide scholarships for UB student-athletes and activities to enhance the visibility and reputation of the university's Division of Athletics.