Taken from the excellent Penalty Shootout Trivia:
In amateur and youth matches probably quite different records have been set; one particular instance went over the world in January 1998: a match for the Derby's Community Cup, a competition for nine and ten years old children, between Mickleover Lightning Blue Sox and Chellaston Boys finished 1-1, and the Blue Sox won the shootout 2-1, but not until 66 penalties had been taken, an "efficiency" of 4,5%.
This means that 66 - 2 - 1 = 63 penalties were missed.
In the "Least Effective Penalty Takers" section of that page we see other cases were 9 penalties were missed since the final score was 2-1 after 12 penalties:
An equally low rate was achieved in the shootout deciding the Finnish Cup final of 1985. It was played between Haka of Valkeakoski and HJK of Helsinki, and had finished 2-2 after extra time. The keepers involved were Huttunen of Haka and Palmroos of HJK, who each got to take a penalty in the sequence of 12. Huttunen made the difference.
The worst shootout in the Copa Libertadores was an equally poor affair. In the 1990 semifinal, Olimpia had lost 2-3 at home to Atlético Nacional after having won the "away" match 2-1 (that first leg was played in Santiago de Chile since the Colombian team were banned from playing at home). The away goals rule not being applied in the Copa Libertadores, and no extra time being played, the teams went straight to the penalty shootout; Nacional's keeper Higuita held 4 out of 6 shots, but his counterpart Almeida emerged victorious, in spite of setting a bad example for his team's penalty takers - like Higuita:
The same rate of 25,0% was "reached" after the final of the South African League Cup in 1998 between Kaizer Chiefs and Mamelodi Sundowns, who had drawn 2-2 after extra time (1-1 after 90 minutes). Kaizer won 2-1 on penalties after 12 spot kicks.