This issue can happen when you have your email accounts (or at least one of them) set up with IMAP protocol.
For IMAP protocol eM Client only synchronizes Message headers by default. So full messages are not downloaded until you open them to view them in eM Client.

This feature is designed to make synchronization more efficient and to save space on your device (for example, if you have thousands of old messages on your server, synchronizing with eM Client will not instantly take over your computer's disk space completely).

When the messages are not downloaded fully in eM Client's database, the bodies of the messages cannot be searched.

If you always want to be able to search your full messages you can enable automatic download of messages for offline use. Simply go to Menu > Tools > Accounts and under the IMAP tab enable the option to 'Download messages for Offline use'. You can also include images and attachments.

POP and EWS accounts download full messages automatically.