I've never played this game online, so no idea about server issues. Does the same problem happen if you play against AIs offline?
Is this a mod track? Does the AI have the same problem with any car? If so it could be a track bug.
Are the AIs revving up the engine while stuck in the pit? If so, that's a common car bug.
Lots of car mods have a bug in data/ai.ini where the UPSHIFT point is higher than the rev limit of the engine. This causes the AI to sit and rev but never put the car in gear.
If that's what it is, just lower the shift points in ai.ini to a range that's appropriate to the car.
Complications:
If the mod has a data.acd file then you need to unpack it first with Content Manager, and then delete it so it doesn't override the /data folder.
After the fix you can repack a new data.acd if you want.
I'm guessing making any edit to the car will cause problems for using the car online though, unless you can get all your users to copy the same updated version that you are using.