Asbestos is a problem word. There are a lot of Codes, both local and State that must be followed when dealing with Asbestos; working with it, disposing of it, etc. This equates to expensive and time consuming work with governmental oversite.
It is best to refer to your siding by the type it is, not as "asbestos siding". Asbestos is a material in your siding, it isn't the siding. You could have cement tile, cement board, masonic tile, etc. Unless you have had the siding tested for asbestos, you are best to opperate under the assumption it contains asbestos, but not declare it as hazardous. You would disclose to your builder / contractor that you suspect asbestos and leave it at that. If you have had the siding tested, then you would give those results to your builder.
That said, removing the siding will be expensive. In VA they require tents to be put up when working with asbestos and lead now. So the contractor would have to put up scaffolding, put up approved tarps or plastic around these scaffolds, provide filtered ventilation, and remove the siding while under tents. Where it is disposed of is also specified, so there will be disposal fees (based upon amount), travel fees (not all dumps will take hazardous buidling materials), etc. The workers will also have clothes covers and face maskes, etc. with filters to be replaced, etc.
Asbestos materials are pretty harmless until disturbed. So the dust from breaking them, drilling, nailing through them, etc is what is dangerous. So when putting up the insulation or siding (you can get insulated vinyl siding BTW), there is a risk when the new material is nailed or screwed through the material. This is why you must let the builder know of your tests if you know it exists (most experienced ones will recognize the age and material and operate as if it is hazardous).
So my recommendation is to cover the existing siding and leave it intact. This will be a faster, cheaper and safer method of dealing with your existing siding.