The meeting organizer MUST use the Microsoft Teams desktop application to utilize the breakout room functionality; however, it is not necessary for attendees to be logged in via the desktop app to participate in breakout rooms.
Microsoft provides a comprehensive guide to using breakout rooms: https://support.microsoft.com/en-us/office/use-breakout-rooms-in-teams-meetings-7de1f48a-da07-466c-a5ab-4ebace28e461