Mailbox Search is by default part of the Discovery Management role.You can assign them via Exchange admin center, Permissions, admin roles (Exchange 2010: ECP, Roles & Auditing, Administrator Roles). The minimum required permissions to be able to perform searches and deletions are:

Deleting emails from multiple mailboxes.Searching and deleting content from one mailbox.I will also discuss the -SearchQuery parameter to manage item scopes and target specific content. If you don’t have anything against using solutions which may stop working, Search-Mailbox can still be of use to you. It’s quick, easy and can delete items in bulk without any problems. While I cannot recommend a solution which is being retired, in February 2022, Search-Mailbox still works miracles. Learn how to use the new version of Search-Mailbox to search and delete mailbox content in Exchange Online, Exchange 2016 and Exchange 2019 Compliance Search cmdlets can only delete 10 items at a time (as compared to 10,000 offered by Search-Mailbox). While the Compliance Search is helpful and doesn’t throw you a “no longer supported” warning, it is less convenient to use and was never meant for bulk content deletion. The alternative to this very useful cmdlet is using *ComplianceSearch cmdlets. Note: In Exchange Online, the cmdlet is no longer officially supported after July 1, 2020.

Outlook 365 delete emails at exit