Anyone can mine resources on the free plot, but the number of workers that can simultaneously work on one deposit is limited. It depends on the server you are playing on: Wild West, Yukon or Grand Land. Also, this limit depends on the resource itself. If the worker extracts on the free lot, the resources are gathered to his backpack. Therefore, a worker cannot obtain more than there's space left in his backpack.
All the resources, except for gold, can be extracted either with bare hands or using the tools. Extraction with tools is much more efficient. One can extract gold with the help of pickax and gold pan only. If this plot is not occupied by anyone, you will see the "Rent" tab on it or you can use the filter and select all unoccupied areas.