Which server are your mailboxes locating on? On-prem Exchange Server or Exchange Online(Office 365)?
A. If it's on-prem server, can you export the .pst of your mailboxes by running the following command in the EMS(Exchange Management Shell)?
New-MailboxExportRequest
a. If the command can't be recognized, perhaps the RBAC permissions haven't worked yet, try running the following command in the PowerShell and then log out/in again to see if there is any differerence:
New-ManagementRoleAssignment -SecurityGroup "Organization Management" -Role "Mailbox Import Export"
Add-RoleGroupMember -Identity "Organization Management" -Member <User>
b. If the issue only happened in EAC, maybe some browser cache caused this issue, I think you could try accessing EAC in incognito mode or using another browser and see if it still exists.
B. If it's Exchange Online, because the command New-MailboxExportRequest which has the same feature with Export to a PST file
is available in only Exchange 2010/2013/2016/2019, the option will not appear even if you assign enough permissions(Mailbox Import Export role or eDiscovery Management Administrator). To export the data of your mailboxes to .pst files, you can try doing an eDiscovery Content Search (Content/Compliance search) and download the .pst file from the search result.