Which organization is appointed for implementing new JS features? For example, adding a new reverse string capability, or anything else that would be introduced in ESNext?
"ES" stands for "ECMA Script". ECMA International is the organization that standardizes Ecma Script. To be more precise, there is Technical Committee 39 or TC39 responsible for this particular standard. Members of the committee are people from different organizations like Microsoft and PayPal. Also there is invited experts, who are not on committee, but also help to develop the language.
They have a process of proposing and introducing features. This process is public and uses GitHub where you can read about what was proposed and propose your own features.
As for implementation - it is responsibility of browser vendors. Most browsers now open sources (at least partially), so you can also contribute.