You do not have to wait for an ESG management standard before you start managing your ESG obligations.
Obligations associated with Environmental Social Governance (ESG) initiatives can cross and overlap several compliance domains.
While standards are being developed unique to ESG with respect to reporting requirements the management of the obligations themselves does not yet have a standard management approach.
For the foreseeable future, deciding where ESG obligations should be managed within an organization and how best to manage them will be largely shaped by the scope and nature of the ESG obligations and the existence of other compliance programs.
For example, organizations that already have EHS programs could incorporate them into those frameworks. You can use ISO 14001 and replace "Environmental" with "ESG" to provide the basics to start moving towards managed ESG.
Other standards that might be helpful to get started:
ISO 37301 - Compliance Management System Standard
ISO 26000 - Social Responsibility (CSR) Management System Standard
Most of these follow a similar framework which can be adapted to include aspects unique to ESG.