Killer Queen Black supports in-game voice chat natively. You do not need the Nintendo Mobile App to use voice chat, and it is not supported or possible to do so.
Alternately, if you are the Killer Queen Black Discord server, and are teaming up, you can use a voice channel instead.
To access voice chat options:
To mute one or more members of your team, press start/+ (Switch) or escape (PC) to pull up a player listing where this is possible.
The simplest approach is to simply plug in an earbud & mic with a 3 ring trrs connector into the headphot jack of your Switch. This works even in docked mode, if you have a long enough cable!
Certain bluetooth headsets now work with the Nintendo Switch. Here is a full list of known Switch-compatible bluetooth headsets.
Unfortunately this is not possible to do currently within Killer Queen Black on the Nintendo Switch.
source @Viotech3#8512 on Discord
If you are teaming up with a group of other players, and therefore won't need to communicate with any "random" teammates through the game itself, many people prefer using Discord's built in voice chat channels.
First, join the Killer Queen Black Discord server, then look in the channel listing in the left-hand sidebar. Under the GENERAL-VOICE-CHANNELS you can see which channels are in use, and which are free.
Agree on one and hop in it with your teammates. You are now up on voice chat.
This works for players on Nintendo Switch as well as PC, simply use the Discord Mobile App and plug in any headphones w/ mic into your phone.