You might possibly then have eg: Categories enabled and the missing emails are in other Inbox Tabs. See the following eM Client Blog.
Now if your Outlook emails online are not in a Category Inbox Tab, then you might have possibly accidentally moved them to another eM Client account mail folder in which case try doing a Search for them in All Folders as per below.
Go to the search field at the top centre, and select “All Folders” in the dropdown field, and then type a search for text that you know is in the Subject or Body of those emails and see if any of those emails then appear.
“eM Client Search documentation”
https://www.emclient.com/webdocumentation/en/10.0/emclient/default.htm#Getting%20Started/Search.htm
