To migrate the user alerts, Sharegate requires that the server extension is installed on the source and destination server. Here is an article how to install it.
SharePoint 2003 and Office 365
The server extension is not compatible with SharePoint 2003. Therefore, user alerts migration from or to a SharePoint 2003 server is not supported. Here is an article that explains what is supported and what is not for each SharePoint version.
Also, to prevent the User alerts from being triggered during the copy, the user must be farm admin at the destination.