Does the Udawalawe entrance fee include the safari jeep?

No. Udawalawe National Park entry charges do not cover a commercial safari jeep. You need to sort out your own vehicle and driver separately, unless you buy a package that clearly says both are included.

Sometimes government admission includes official vehicle entry fees and other required charges. That is separate from what the safari operator charges for the jeep rental. Before you book, check that the safari quote covers the park tickets, the vehicle, the driver, and any government taxes that apply.