The connection to the API and to the integrations with code hosting platforms (GitHub, Bitbucket, GitLab, Azure DevOps) is set at an account level. For this reason, the owner of the projects is usually the one does the language file management when these automation tools are involved in the localization process.
But an administrator assigned to a project can also take over this task.
To set up the API and import/export strings, they can just use an API key generated in their own POEditor account account.
To work with the GitHub, Bitbucket, GitLab or Azure DevOps integrations, admins can use Webhooks.
First, the project owner should set the links between the language file in your repo and the corresponding language in POEditor. After this, anyone can trigger the updates both ways: to get the terms (and translations) from the repos and to export the translations in POEditor to the linked file in your repos.