Carbon emission reduction is increasingly becoming a public consensus, with governments formulating carbon emission policies, enterprises investing in emission abatement equipment, and consumers having a low-carbon preference. On the other hand, it is difficult for industry managers to obtain all the demand information. Based on this, this paper aims to investigate operations and coordination for a sustainable system with a flexible cap-and-trade policy and limited demand information. Newsvendor and distribution-free newsvendor models are formulated to show the validity of limited information. Stackelberg game is exploited to derive optimal abatement and order quantity solutions under centralized and decentralized systems. The revenue-sharing and two-part tariff contracts are then proposed to coordinate the decentralized system with limited demand information. Numerical analyses complement the theoretical results. We list some major findings. Firstly, we discover that using abatement equipment can effectively reduce emissions and increase profits. Secondly, the distribution-free approach is effective and acceptable for a system where only mean and variance information is informed. Thirdly, the mean parameter has a greater impact on profits and emissions comparing with the other seven parameters. Finally, we show that both contracts may achieve perfect coordination, and the two-part tariff contract is more robust.