How do I get the source code of the submission to lay out properly? I cannot get the markdown syntax for code blocks to work.

Are you referring to Itch.io's game page editor?
If so, I just use the (¶ Format) > "Code" option from the toolbar of the main text editor.

If you mean within the actual jam submission form?
Then I don't think that's possible, as it's a simple multi-line text box - so it'll just show the raw text.
(Which is fine, as it's intended for me/others to be able to copy + paste your code to verify the entry, if needed)

Hope this helps 🤓

Yes, I meant in the jam submission page. Thanks for the explanation.