First of all, a warning. The problem you're trying to solve is complex enough that most professional games don't even bother unless it's a key part of the gameplay. So my advice would be to cheat as much as possible.
For vehicle radius, I would try to use the worst case (the largest dimension) and then treat the vehicle as a cylinder (or a circle, in 2D) for pathfinding. With navmeshes, you can either have a different set of meshes that take into account the different radii, or make the pathfinder stay away from the wall a certain distance. In either case, picture pushing the edges of the navmesh inwards and rounding the corners (this is technically called a Minkowski Sum). Then you can treat the vehicle as a point, in this expanded navmesh.
If you want to take into account turning, you'll have to encode those rules in the pathfinder itself, as it expands. For instance, if a vehicle can't turn in a tight radius, you'll have to both discard some of the polygons offered as connections from the current one, and use this information when optimizing the path during string pulling. In general, you'll also want to encode some of this logic in the heuristic, so that paths try to minimize turns and look as natural as possible.