Hey castorlight,
Most symphony orchestras have regular rehearsals. Perhaps, if you met and spoke with the conductor, he'd let you use them during one of the rehearsal times to lay down a music bed for your piece. Barter the fact that they'll get publicity when your video is released. It's worth a shot.
The other option is to pick up an instrument and actually join a community symphony. Make friends, talk to the conductor, etc...
There's no such thing as a free lunch. If you want symphonic music that's not royaly free synth garbage, then you're going to have to come up with something that will get the conductor's attention and make it worth their while. If you're looking to have an original piece composed altogether, just pay them to do it.