Is it possible to create multiple org chart web parts with different data sources? We have few branches and I need to create separate org chart for each branch.
Yes, you can create as much web parts as you need and configure data source for each of them separately.
If you use SharePoint lists as data sources, you can create separate lists for each web part.
If you use User Profile Service as data source, you can specify person, from which org chart will be displayed (root person).
For more information about the Web Part configuration read documentation: http://plumsail.com/sharepoint-orgchart/docs/advanced-web-part-configuration/