Short answer: not always, but it helps a lot. It depends on where you go.
In backpacker hubs like Bundaberg (QLD) or Coffs Harbour (NSW), 'working hostels' usually arrange the job and even drive you to the farm in a van. You can manage without a car there.
In more isolated areas like the Riverland (SA) or much of Western Australia (WA), farms are spread out and public transport is scarce: there a car is practically necessary to land and keep the work.
Perks of having a car: you reach more farms, switch jobs fast when a crop ends, and negotiate better because you don't depend on the employer's transport.
Tip: if you've just arrived with no car, start in a hub with working hostels and, once you save some money, consider buying a used car for the next seasons.