Is there any way, where I can use a API to load all organizations a given user is member of?
The REST API endpoint
Returns a user's course and organisation memberships.
It won't tell you which are the organisation ones and there isn't a filter option for this, so you will need to make another call using the course id to see if the membership is for a course or an organisation, unless you have some other way particular to your data. For example, if you can tell them apart using the dataSourceId, or if they have a custom role.
Another possibility might be to make the API call with a user who only has the system.user.org.enrollment.VIEW entitlement and not the system.user.course.enrollment.VIEW entitlement.
Retrieving data ...