For the OP, no one that I know of will pick up your car on launch day and then keep it locked up until delivering it on your take-out day. The roughnecks and associated culture that have swarmed into Vernal for the gas fields have made the Split Mountain takeout a little higher-risk than it used to be. Swasey's seems pretty safe still. If you fly, you'll need to get to the airport, which is actually south of Green River by a little bit, and requires another shuttle just to pick up all your vehicles.
I know River Runners has gone above and beyond for many people, but the drivers have also "confused" gas money and tip money and left a friend's car at Swasey's running on fumes 2 years in a row, and have pushed my truck up to higher RPMs empty than it took me to get over from Denver fully loaded, and then told me I was lying when I called to give them feedback. (not complain, just inform so they could talk to their drivers). My take-away is that if you aren't driving your own car yourself, it's a crap shoot.