Indeed, Canvas Inbox doesn't have a keyword search feature, even though it has been requested years ago, seemingly multiple times. See, for example, [Inbox] Add search feature for Inbox/Conversations - Instructure Community Ideas.
Alternatively, it's possible to create a simple (and probably slow) search feature using the API, by looping through the conversation IDs returned in /api/v1/conversations and then getting the body of each message in /api/v1/conversations/:id, but I agree that could have been available in the UI.
See Conversations - Canvas LMS REST API Documentation for more details.