I use Aseprite for pixel art, which is paid but you can also compile it for free, and I leave the .aseprite images on my asset folder. Do I need to delete them?
I might be wrong as I'm not part of any ruling party in this contest, but as explained in the "Reminder" post (https://itch.io/jam/gamedevtv-jam-2022/topic/2055595/reminder-all-submissions-mu...)
You have to includes all the sources files (including the raw build folder of your project) with your submission, but it doesn't have to be WITH the game's playable file. You can include the source files as a separate URL such as a Google Drive, Dropbox or even a direct download link of file you may host on your own website. (You submit your game as a Build and, in the details, a link to download the source files.)
So, in your case, you can just put the generated PNG files from your Aseprite file and include your source .aseprite image in your "Source" files.
What's not allowed is to use paid plugin and tools within the selected engine as you got to remove any paid assets or tools from your source file before submitting it. The example given is Gaia which is a Terrain tool that allows you to generate mesh and textures for a 3D terrain, but also advanced terrain (including vegetation and other stuff) directly in the engine. Someone can use Gaia to generate a FBX terrain mesh, then use that FBX file in their project, but they can't use a Gaia terrain in their project as they will have to remove Gaia and this will break the source files.
An example of the Aseprite files would be to include your .aseprite files in a source folder which will be submitted with the project folder in the Dropbox or Github or whatever file storage URL you'll be giving, but make sure that the project folder can be opened and edited without those .aseprite files unless Unity can open it natively or with a FREE plugin.
For example, you can include the files of the Animation Importer Tool by Talecrafter (https://github.com/talecrafter/AnimationImporter/releases) as this is a free plugin available to all without paying anything to import PNG files generated from Aseprite in your project folder, but you couldn't use the Asprite Importer tool by Rodrigo (https://assetstore.unity.com/packages/tools/sprite-management/aseprite-importer-...) to import your .aseprite image as this tool is paid (10$) and generates codes and files that are required to use the resulted files in the project.
Another example of a tool that you (or I) can't use would be the Toony Colors Pro 2 tool since it's a paid asset that generate custom Shaders within Unity. I usually use that tool to generate the bottom base of my shaders since I like how it generate such a clean and easy-to-work-with shader infrastructure, but I can't use this tool because it generate the 1/3 of the final shader codes for me. I have to either write my own shader from scratch or use and edit common public shader base codes instead like the ones you may get from the few shader tutorial around.