I would wait until you are ready to make the move. Then get your own IRP plate for the state your business is located. I have no issues with NY other then some Insurance companies not giving insurance.
Unless you plan on doing loads outside the company you lease to, it may not be worth getting your authority.
The answer to your question is no. It is their authority and plate, you need your own liability insurance & plate to apply for your own authority.
The added expense of insurance for own authority is one of the bigger expenses, along with fuel and truck payment. By being leased onto a company you are saving about $1000 a month in insurnance and can save on not paying fuel taxes and registration fees.
I can use power tools, and I'll think about doing your dry cleaning if the rates are good.