In most places starting up a simple lawn care business does not require any license and you can easily start mowing lawns for people without a business license.
But forming an L.L.C and getting a business license will help with your lawn care business because it shows people you're serious about your lawn care business and not just some person looking to make a quick buck.
The business license also shows you're more professional and will get you more business as more people and businesses will trust you with the lawn care and will be more likely to hire you when you do have the business license.
Getting a business license is pretty easy and not all that expensive so I would recommend getting the business license if possible.