Several GitHub repositories have ported Team Fortress 1.5 (the Quake mod) to run in HTML5. Search for "QWTF Web port." While it isn't TF2 , it has the same classes (Scout, Soldier, Medic) and the same frantic pacing.
Legitimate ways to play TF2 when access is limited ---- Team Fortress 2 Unblocked No Flash
GeForce NOW is the undisputed champion of "Team Fortress 2 Unblocked No Flash." It requires no download, no Flash, and runs on a potato laptop. Several GitHub repositories have ported Team Fortress 1