Removal of Asbestos siding can cost as much as the the new siding job. Depending on the State that you live in, the cost will vary. Abestos siding is a hazardous material and must be disposed of properly.
Your Home Improvement Co. should know "up front" what the cost will be.
You may want to find out if you can still go over it in your area. If you choose this option, you want to have "atleast" 3/8" fanfold insulation installed over the asbestos to keep any material that may chip or break in the future from making your vinyl siding buldge.
I would vote for removal of the asbestos if at all possible because it will have a affect on your re-sale of the property.(you would have to disclose it.)
Also, the asbestos was probably installed over wooden clapboards. If you remove the asbestos, you should have room to install a better insulation then the vinyl siding, without building the wall out too much, making your windows look like they are in tunnels.