Long Arm quilters do not normally do the binding on a quilt. If they do, they charge extra. Some will run it down the front side for you but that seems to be rare.
I'm not sure what you mean by binding with the Chevy Logo. Do you have fabric with the logo on it are do you plan on stitching it down by machine on the front?
If you mean the border, yes, LAQ's can by free hand, computerized or stencil, quilt the Chevy bowtie design.