The problem is, they only support once, then disembark and won't do anything else.
If you as a player have used their support, you need to call "Support done" after they're done. That way, they will be released back to the support system, so to speak.
The "support done" call is made by pressing 5 - 1 - 5 (Status - Call Support - Done)
Now, the bad news: the support system doesn't work properly in MP. Bugged. Support units will not heed calls for support from players unless the supporting unit is local to the calling player's machine.
That means it will not work for any player if the mission runs on a dedicated server. AI:s not under player control however, are able to call for support. They, on the other hand, however, have the problem of never calling "Support Done" so supporting units will most often remain on the front line.
It's funny how most things in OFP related to MP beyond the bare neccessities for TDM or other simple mission types are broken...