1. Is it possible to use local Veeam server to take backup on Exchange server online and how it works. I checked the page and I can see that there is no support for Sharepoint yet.
Office 365 mailboxes can be backed up via Veeam Backup for Microsoft Office 365. Support for Sharepoint is coming as well (no ETA at the moment, though).
2. How to take backup on VMs in Azure? Can we use free Veeam agent for Windows? Can we save backup from Azure VMs locally?
Correct, you can backup Azure VMs using one of our agent solutions. Such backups can be saved either in Azure or in your production environment, assuming there is a repository server running locally and it can be accessed from Azure VM.
3. Can we use Veeam agent for windows free version to take backup on client computers and send them to our veeam server?
Veeam agents can use backup repository as their target without any issues, if that's what you're asking for.