I want a few users who use a shared mailbox to be able to send mails on behalf of a user. This option is not available in the exchange administrator tool. How can this be resolved? Would it be an option to give this mailbox an exchange online account?
You can have the email of the represented user to show up in Outlook automatically to send and receive.
Go to Office 365 Portal, go to the Admin area, then edit users, then select who you want someone ELSE to email as them, click on the "Mail Settings" towards the bottom, click Edit on mailbox permissions. Click on "Add permissions" button, then you'll want to add the user who's sending the emails to "Read and Manage" and additionally add them to either "Send on behalf" or "Send As". These are represented different when the receiver gets them. For example, a secretary would want to be able to "Send on behalf" of a manager, but the secretary would want to "Send As" the general info mailbox.
After a short period of time (seen as little as a minute, as long as 30 minutes) you'll see the additional mailbox automatically propagate in the emailing user's Outlook. Now the trick is having the email appear from the user, in a new email window click on on the Options ribbon tab, then click on the "From" button to make the from field visible so you can change it if necessary.