It depends entirely on the security. Sometimes users will pick far out of the money strike with little volume and be the entire market for that day, but major stocks like AMZN have such deep option pools and liquidity that even the entire sub buying the same thing would be insignificant.

Options are derivatives of the security so any influence on share price will be a very diluted reaction based on risk and general market sentiment which I don't think a few thousand users are capable of. They would need to carry serious notional volume (in the 9+ figures) or major hedging risk with their brokers for any price action to occur, and even that might not last beyond days or hours.