Hi Krisztian,
Based on my experience, there are two ways to meet your requirement.
The first way, you can use PowerShell to manage access permissions for users’ mailboxes. To do this, please perform the steps below:
1. Use Windows PowerShell in Exchange Online to connect to your service. For detailed infoamation, you can refer to article: Use Windows PowerShell in Exchange Online
2. Run the following command to grant a user permission to access a folder in another user's mailbox. This example assigns permissions for Ed to access Ayla's Marketing mailbox folder and applies the Owner role to his access of that folder.
Example: Add-MailboxFolderPermission -Identity ayla@contoso.com:\Marketing -User Ed@contoso.com -AccessRights Owner
The second way, to create a folder to share emails in Office 365, we recommend you use Shared Mailbox. A Shared Mailbox is a mailbox that multiple users can open to read and send e-mail messages. Shared mailboxes allow a group of users to view and send e-mail from a common mailbox. This online article introducing detailed steps is for your reference. Set Up a Shared Mailbox
Thanks,
Maggie Li