Yes, there are actually lots of eco-conscious UK Fulfillment Companies. The entire question of whether or not a fulfillment service is eco conscious lies in the doctrine and practices that they adopt.
Therefore, to identify such a company, you have to be on the lookout for such practices. Currently, they include sustainable packaging (which is the use of eco-friendly and recyclable package materials), efficient routing to optimise fulfilment and reduce transportation emissions, and energy-efficient warehousing.
Other important practices include minimising excess packaging and reducing waste through smart packaging practice, carbon offsetting to mitigate environmental impact, and green transportation such as electric vehicles.
Granted, not every eco-conscious fulfilment company will be able to afford all of these activities, which is why eco-friendliness is a spectrum. All you can do is go with a service that adopts as many of these practices as possible, and you will be supporting an eco-friendly company.
