Stuppacher was born in Vienna in 1947 into a family of means — his mother, Helene Musa, had inherited one of the largest property management companies in Austria. His sister, Elisabeth Stuppacher, competed in harness racing and was the Austrian champion in that discipline from 1993 to 2000.
He began racing in 1968 in a Mini Cooper, then rapidly acquired a Porsche 906 for Austrian national sportscar events. By 1969 he was competing internationally. Racing with the Bosch Racing Team under the shared entry of Kurt Rieder, he finished second in class and twelfth overall at the 1000 km of Monza in a Porsche 906. That July he placed eighth in the inaugural race at the newly opened Österreichring in a Porsche 910, and in August he shared a Porsche 910 with a young Niki Lauda at the 1000 km of Zeltweg, finishing 21st. He also took a class win at the Preis von Tirol hillclimb in Innsbruck in the Porsche 910 that year.
In 1970 Stuppacher won an outright victory on the Těrlicko street circuit in Czechoslovakia in his Porsche 910. His most significant career success followed in 1971, when he purchased a Porsche 908/02 spyder from Lauda and won the Austrian Hillclimb Championship.
His 1972 sportscar season included a ninth-place finish at the 1000 km of Brands Hatch sharing the 908/02 with Rieder. He made his 24 Hours of Le Mans debut that year, sharing the car with Walter Roser; they qualified 21st but retired after approximately two hours following a crash. A two-year hiatus from 1973 ended with a single entry at the 1975 Zeltweg World Championship round in a Lola T294-Ford entered by Roger Heavens Racing.
In 1976 the ÖASC Racing Team purchased a Tyrrell 007-Ford Cosworth — the former car of Jody Scheckter — and a March 761-Ford Cosworth, entering Stuppacher and teammate Karl Oppitzhauser for a programme of selected Grands Prix. Both drivers were refused entry to the Austrian Grand Prix at the Österreichring on grounds of insufficient racing experience; attempts to secure support from other teams via a petition to the paddock failed. Oppitzhauser ended his Formula One ambitions at that point.
Stuppacher continued to the Italian Grand Prix at Monza, starting 29th and last in qualifying — nearly 14 seconds from pole-sitter Jacques Laffite's time — and departed for Austria immediately after Saturday qualifying, satisfied that he had not made the 26-car grid. In his absence, the race stewards excluded the cars of James Hunt, Jochen Mass, and John Watson for fuel irregularities, promoting Stuppacher into the field; however, he was already back in Austria and could not return in time to start. Hunt and Mass were subsequently reinstated after Arturo Merzario and Guy Edwards withdrew their entries, leaving Stuppacher without a start despite the brief promotion.
The team then travelled to Canada for the Grand Prix at Mosport Park. Stuppacher was around eight seconds slower in qualifying than the slowest car that did make the grid; other drivers collectively requested he withdraw, but he did not comply.
At Watkins Glen one week later, in wet qualifying conditions, Stuppacher posted a time 27.448 seconds slower than James Hunt's pole lap. That gap remains an official Formula One qualifying record. The Tyrrell 007 he drove carried "Austria is beautiful" tourism advertising on its nose cone, the space funded by his mother's property business. With that attempt his Formula One campaign concluded.
After 1976 Stuppacher occasionally competed in European hillclimbs in a GRD 274-BMW Formula Two car and made a brief return in the Alfa Romeo Alfasud European Cup in 1981. He sold his Tyrrell 007 in 1985. Very little is known about his life in subsequent years. He was found dead in his Vienna apartment on 13 August 2001, aged 54.