Lahugala isn’t officially open to the public, and no vehicles are allowed in, although you’re free to walk into the park from the main road between Arugam Bay and Monaragala, which runs right through it. Be aware, however, that walking through jungle with a large elephant population carries a degree of risk, so it’s best to stick to one of the recognized viewpoints close to the road. The easiest (and safest) option is to head to Lahugala Hospital (at the 306km post). Just west of here along the main road, several small paths run off to the right to the raised bund at the edge of Lahugala Tank, about 100m away, which offers a secure vantage point and good chances of spotting elephants.
Guesthouses in Arugam Bay can arrange jeeps to Lahugala, although given that you can’t actually drive these into the park, you might as well save your money and catch a bus or tuktuk.