We fixed the problem with the update button in Automatic Email Manager since version 9.03.1108.
You have to download and install manually this latest built (all your data is saved during update), make sure to quit Automatic Email Manager before running installer:
https://www.automatic-email-manager.com/download/automaticemailmanager.exe
or from the official page:
https://www.automatic-email-manager.com/download-automatic-email-manager