Even assuming that in the JAG universe Space Shuttle missions are flown from Vandenberg AFB (in real life plans to do this were scrapped following the "Challenger" accident in 1986), such a launch could never reach a satellite destined for geostationary orbit. Vandenberg sends satellites into polar orbit, launching them in a southerly direction; launching to the east, which would be needed to match orbits with the target satellite, would take the Shuttle over inhabited land during the ascent, which is prohibited.
When Rabb suggests that they get more people out of the Shuttle to manually capture the satellite, the mission clock in the background read 3 hours, 23 minutes and change. When the capture is successful, the mission clock reads 3 hours, 24 minutes and change. It would be impossible for astronauts to put on their EVA suits, conduct the systems checks, exit the shuttle and capture the satellite in 1 minute.
At about the 25 minute mark a graphic shows the countdown at T-minus 1 hr 12 min to launch, with the crew yet to board the shuttle. Shuttle crews typically boarded the craft about 2.5 hours before scheduled launch time.
The Space Shuttle Endeavor is shown being launched from Vandenberg Air Force Base, which for a time was an alternate launch and landing site for the space shuttle program, but it was never used for that purpose. Following the Challenger disaster and due to budget cuts to NASA Vandenberg was removed from the space shuttle program in December of 1989, six years before this episode takes place.
When the Endeavour is approaching the satellite the crew is shown wearing regular civilian clothing: polo shirts and jeans. Shuttle crews wore flight suits at all times, especially military crew members.