I suspect most Vinyl cutters will come with their own software. I am absolutely not sure but I would think that Illustrator could output to a cutter - given the right driver. I realise that you would need to manually work out the viability of a design.
Fredfish is right. Usually, drivers are installed on a specific model of the cutter. If you are not sure how to install the drivers or you do not know the model of your cutter, look for information in vinyl cutting machine reviews, or contact the store where you purchased your machine.