No, not everybody had a tractor - that would be ineffective, given how many small family-owned farms there were. But they did use tractors and combine harvesters. TBH I don't know whether they were shared between neighbors or just unofficially borrowed(#) for a bottle of vodka from the state farms (PGR).
(#)There's this Polish verb "załatwić" which I cannot properly translate to English. It means obtaining products or services using ones connections, cunning and possibly transfer of a usual token of appreciation as in "załatwić za flachę" (obtain something in exchange for a bottle of vodka).