Is there a way to get there from Manhattan without a car?

You could take the LIRR to the Syosset train station and take a cab to the Planting Field as there is a cab company right there. Also it's about 15 mins by car from the station.

You could try uber from a train station. Or possibly rent a car for the day.

Do not think so

I checked using Google maps, and it says it can't find a way with public transport. That its 50 minutes by car/cab, 10 hrs walking!!! Or 3 hrs by bicycle! Sorey I can't help further.