I'm not too familiar with how Flatpak works but Emacs benefits from compiling it on your machine natively. Tell me what distro you're on and I can see if I can find out how you'd do that.

Following up from my previous comment, there is a Flatpak of Emacs available on Flathub. Here are the instructions for how to install, whilst enabling native compilation, which will offer a performance increase and allow you to use features such as vterm (the best terminal emulator for Emacs).

A Flatpak of Doom Emacs? No. But you can just install the normal Emacs flatpak and then install Doom Emacs with 2 simple commands:

git clone --depth 1 https://github.com/doomemacs/doomemacs ~/.config/emacs

~/.config/emacs/bin/doom install

Emacs will read these config files from the .config/emacs directory. Doom Emacs is not a different version of the program, it's essentially just a set of configuration files.