Blackhand
I have just searched the FAA STC database and similarly had trouble find any STC's. I have searched for GAMI under STC holder and I have looked specifically at a number of engines listed in the GAMI compatibility chart and a number of aircraft mentioned as suitable for fitting GAMI injectors.
The GAMI website makes mention of them holding STC's, but I cannot find any STC numbers.
Ultimately, I found:
STC 09289SC for Continental IO550 B, BA & BB
STC 09963SC for Continental IO 360 ES, R , G & N
STC 09217SC for Continental IO 470 series and IO 550 series
STC 09401SC for Continental IO 360 series
STC 09445SC for some Lycoming engines
The FAA does not provide any text of approval documentation.