For our company's chat we are using Skype for Business. The sidebar is currently accessible from the Outlook web app:
Unfortunately as of this moment, you can't customize the Suite bar to show the SfB button.
The only option you have is OneDrive for Business and Sites.
If you have Office 365 Education subscription, then you can show OneNote Class Notebooks and OneNote staff notebooks.
Below is the screenshot of the SharePoint settings page.
Login to this site using Tenant/SharePoint admin credentials to control the settings.