Hi David and Lizzy,
There's currently no way to turn off chat in the lobby - however, the lobby is just an intermediary step that your child goes through to access their private server.
When your child joins the Tynker server, they'll be placed in the lobby, but as soon as they type "/join", they'll be taken to their private server. To test out mods and resource packs, students must be in their private server, not the lobby.
No one can join your child's private server unless your child has whitelisted them by username.
Please let us know if this didn't fully answer your question or if you have any additional concerns.