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If you drive a company vehicle you absolutely can ship it to the location of your new position. In most cases you will need authorization from your employer once you have the auto shipping arranged as they will want to be sure you are working with a company that has sufficient insurance and proper licensing.
The company will in most cases reimburse or directly pay for shipping a their vehicle as you will need to have transportation to perform your tasks that pertain to your job.
There are exceptions, however:
It is best to clarify with your human resources department what is covered during your relocation, and what is not. If the relocation is not far enough for shipping to be covered, remember to ask them if they allow you to transport their company car if driving is something you insist on not doing.
If you are concerned with shipping your company owned vehicle to your new location, it is possible your company could have preexisting arrangement with different movers or an auto shipping company. This can eliminate a lot of planning on your part if they handle everything.
In some cases the shipping agent or carrier used by your company could have delays or not meed your timeline. In those cases you can always ask if they will allow you to chose the company should their preferred shipper is having trouble with your transport.
Of course being your personally owned vehicle that you have bought and pay for, you will have a choice in regards to who will load your vehicle. Therefore you are free to chose who you want to take it for you.
Even if your company has a team that handles shipping company owned vehicles, they may not actually want to ship your personal car for liability reasons. In which case with most relocations, they will reimburse you as they may already do so for expenses like tolls, maintenance, and even wear and tear.
Many corporations will opt to pay for your moving and even closing costs when buying house (if your position justifies it). If you are a valued asset you will be surprised what is covered and it may be just a matter of putting in request. So they will likely cover your personal vehicles and at times even those of immediate family.