I think this is going to depend on the station and how parked up the garage is. In the locations I've rented from in Manhattan, the cars are driven out to you from their storage space, so I assume they are double- or tripple-parked in the garage and they have to move cars around. If you want to do what you propose, the station is going to have to be willing to park the car in a part of the garage where it doesn't block in any other of their cars (assuming you are wanting to take the keys with you and lock your bag in the trunk).

Over at Jersey City, where the station is within a large office building parking garage, they give you the car keys and a validated parking ticket and direct you to retrieve your car upstairs, so I've never had a problem checking out the car and getting the contract/keys/ticket during my lunch hour and then driving off after work (but note that the contract starts when you do this).

For NYC, I think it will depend on the station; I would call in advance. I suspect the location I frequent the most (12th St.) would do this, as they know me pretty well.