Below are some contributors to responsible technology practice who work specifically on developing methods and tools for technology developers.
Of course there are many more working on other aspects of ethical technology, from policy, law, and philosophy to tackling specific issues such as privacy and transparency. For this broader view, see this ecosystem map.
Doteveryone - A think tank championing responsible tech for a fairer future. Their TechTransformed initiative focuses specifically on methods for responsible innovation.
IEEE - The largest professional engineering organisation globally, the IEEE "Ethics in Action" initiative has seen the release of a large cooperative report on Ethically-aligned Design" and the start of a set of new standards projects (IEEE P7000) for ethical intelligent and autonomous technologies.
Orbit - "ORBIT provides training, consultancy and other Responsible Research and Innovation (RRI) services to both university and industry research and innovation teams to help ensure that RRI principles are embedded in development from the earliest possible point."
Positive Computing - An initiative for helping technology makers design in ways that respect psychological needs and foster wellbeing (wellbeing-supportive design).
Please let us know of other groups that should be on the list.