Notes:
It looks like they use Amazon Chime as for handling the actual calling stuff. I have yet to try this, but you might be able to use the regular browser Amazon Chime SDK for joining calls
Unless otherwise denoted, these all require a Slack client token for authentication. These are different from user or bot tokens (the kind you get from api.slack.com) - client tokens start with xoxc- and can be found by looking through your browser's cookies/recent HTTP requests. It is worth being careful with this token since it authorizes anyone with this token to do everything you can do in your Slack app, including sending/receiving messages.
Most of the interesting stuff is in Join Huddle
I noticed a few events being sent over the main WebSocket connection that Slack uses for general client functionality. I need to try these again with other people and actual audio to see more of what the responses are like, and I'll eventually remove the IDs and event timestamps from these, but there's nothing super sensitive here for the most part.
{"type":"huddle_invite_cancel","team_id":"T0266FRGM","channel_id":"C0266FRGV","event_ts":"1628135021.355900"}
{"type":"sh_room_join","user":"UN6C43287","room":{"id":"R02AYBUK84Q","name":null,"media_server":"","created_by":"UN6C43287","date_start":1628135005,"date_end":0,"participants":["UN6C43287"],"participant_history":["UN6C43287"],"participants_camera_on":[],"participants_camera_off":[],"participants_screenshare_on":[],"participants_screenshare_off":[],"channels":[],"is_dm_call":true,"was_rejected":false,"was_missed":false,"was_accepted":false,"has_ended":false,"media_backend_type":"free_willy","external_unique_id":"<CHIME_CALL_ID>","app_id":"A00"},"huddle":{"channel_id":"C0266FRGV"},"event_ts":"1628135021.142400","ts":"1628135021.142400"}