Thanks @CrashyBang and @KaMiila, your solutions worked wonderfully for my similar issue with Discord Flatpak. I'll detail my exact procedure for anyone else:
Step 1: Add the following lines to ~/.xinitrc (before the line that starts WM)
mkdir -p "$HOME/.cache/xdg_runtime_dir"
export XDG_RUNTIME_DIR="$HOME/.cache/xdg_runtime_dir"
Step 2: Create a script (mine is ~/.local/bin/Discord) with the following lines
#!/bin/sh
export xprop -root -notype PULSE_SERVER | tr -d ' '
PULSE_SERVER=unix:/tmp/pulse-PKdhtXMmr18n/native
flatpak run com.discordapp.Discord
Step 3: Change the permission on the script to make it runnable
chmod a+x ~/.local/bin/Discord
Step 4: Restart/reload your WM and run the script. If Discord still has no sound the first time you run it after this procedure, kill all Discord processes (check htop) and try again. Sometimes xdg_runtime_dir is not populated before discord launches so it can't connect to PulseAudio but after restarting Discord, it should work.