Assuming there are no legal reasons in your state that keep you from doing so, you can definitely start the business before you get your license and this is probably a good idea. There are lots of things you need to do before you start doing jobs and getting your license is just one of them.
You should immediately determine the form of business you are going to have (sole proprietor, LLC, corp, etc), determine the name you will be using, and then register it with the state. You should determine whether a good internet URL is available when determining your name. If you are going to be something other than a sole proprietor, you'll need a registered agent to handle your official document deliveries.
The sooner you form the company, the sooner you can start using the company name to shop for vehicles, insurance, lawyers, bookkeepers, materials, credit apps at vendors, office space, payroll services, web services, trademarks, loans, bank accounts, business credit cards, and 100 other things. Things like checking accounts and credit cards need your company name before they can be issued. You will want to write company checks to cover start-up costs like licence exams and prep materials.
People/vendors will take you more seriously if you have a registered business, plus you don't want to start buying stuff in your own name and then have to transfer the stuff to the business. That's messy accounting wise. If you're allowed to do certain work without a licence in your jurisdiction, having a company will make you appear more trustworthy to potential clients.
