A native of Toledo, Ohio, Snyder played three years of football at Ohio University, seeing action on both sides of the ball. During that trio of seasons that began in 1933, Snyder was off the field for just four minutes, winning small college All-America honors as a senior.
On February 13, 1939, Snyder was traded to the Chicago Bears and used at running back. After contributing throughout the campaign, he suffered a deep personal tragedy when his two-week-old son, Robert Calvin, died on November 30.
When he returned the next year, Snyder played exclusively at quarterback (and continuing as a kicker), but had the misfortune of playing on the same team as future Hall of Fame signal caller Sid Luckman. That frustration was soothed somewhat when the Bears went on to capture the 1940 NFL Championship with the biggest rout in the league's postseason history, defeating the Washington Redskins, 73–0.
After another season of play resulted in another championship for the Bears, with Snyder kicking three field goals in the 37-9 championship game victory, Snyder retired to become the University of Notre Dame's freshman football coach on July 10, 1942. His main focus was instruction in his area of expertise, the T-Formation, and his work undoubtedly proved to be successful when two quarterbacks under his tutelage, Angelo Bertelli and Johnny Lujack, went on to win the Heisman Trophy.
Snyder returned to the playing field with one more season with the Bears in 1943, notching a trio of league kicking records with 39 conversions, while also connecting on all 12 of his field goal attempts. During the November 14 contest against the New York Giants, Snyder scored on eight conversions during the Bears' high-scoring effort. After the season ended with another Bears' championship, he spent one year contributing to the war effort, serving as personnel manager of the Thompson Aircraft plant in his hometown of Toledo. A back injury suffered during his playing days had prevented Snyder from serving in the armed services. During this stretch, he also coached Libby High School, his alma mater, to an Ohio state title.
In 1945, Snyder joined the Rams as an assistant coach, and played a major role in helping rookie quarterback Bob Waterfield win Most Valuable Player accolades. Under bitterly cold conditions on December 16, the Rams won the NFL title with a 15–14 decision over the Redskins. Just weeks later, Ram owner Dan Reeves shifted the franchise to Los Angeles, with Snyder spending the following season in the same capacity.
After Rams head coach Adam Walsh resigned at the conclusion of the 1946 NFL season, Snyder was named head coach on January 20, 1947 and immediately became the league's youngest head coach. Winning three of the first four games of the regular season, Snyder watched the Rams collapse at midseason with a four-game losing streak to finish with a 6–6 mark. The pressures of that season followed him into the next, causing him to resign on September 3, 1948, citing an ulcer condition.
Despite that stress, Snyder would stay away from coaching for less than a month before accepting an assistant post with the USC Trojans. On January 28, 1949, he left that position for an assistant coaching spot with the Green Bay Packers under Curly Lambeau, but again remained only a year before heading home to become head football coach at Toledo.
Snyder made a bit of history by naming Dick Huston as freshman coach, the first African-American mentor to ever work for a non-black college. Despite being back among family, Snyder left the school after finishing 4–5 during the 1950 season, and wouldn't resurface until 1953 when he was named head coach of the Canadian Football League's Calgary Stampeders.
That foray would again only span one season before he was dismissed. The aftermath of his departure was ugly as Snyder claimed in February 1954 that players were taking the amphetaminebenzedrine to keep up their stamina during the season, a charge that was roundly denied by the league. In August of that year, Snyder spent one season as an assistant at Villanova University, then returned to the NFL for two years with the Pittsburgh Steelers before moving back for one year to the college level as a West Virginia University assistant.
Snyder watched his coaching career wind down during the 1960s, working at the minor league football level, beginning with a stint with the Toledo Tornandoes of the United Football League. On September 19, 1965, he was named head coach of the Wheeling Ironmen of the Continental Football League, lasting until his resignation on November 22, 1967. The following year, he became head coach of the league's Indianapolis Capitols, but resigned on February 4, 1969 to enter private business in Toledo.