I had the same question. First I tried using adb shell to put a symbolic link to /storage/emulated/0/Documents/TIC-80 (my folder of carts I copied over) into /storage/emulated/0/Android/data/com.nesbox.tic/files/TIC-80/ but only the application has access to that folder. That also means you can't copy files to that folder while connected to your computer.
You could get your code copied over by opening the cart you've copied to your device in a text editor app, copying the code text, and then paste it with CTRL+V in the TIC-80 editor.
The app should have an Android intent that lets other apps, such as a file manager or the web browser after download, share a .tic file to it so you can open files that way. Not sure if the app also needs file system permission with that or if the intent system just passes the object without necessitating that (probably?) Or the app could have file system permission and create the TIC-80 folder at /storage/emulated/0/TIC-80 instead. That way users can copy files from another machine and play them.
Can't remember if mentions work on itch but worth a shot. @Nesbox, can you review my suggestion?