If the site maintainer is interested, I have a fairly automated way of pulling down LAT puzzles and converting them to PUZ file based on Android shortyz app code in GitHub. I can help set this up if needed.
I'm one of those two that jmh123 mentioned. Like was said, "site maintainer" hasn't talked any about the issue. I may have to directly ask. I'm guessing based on when PUZ files don't post here that there's some kind of manual intervention required on Mondays and the end of the month that doesn't always happen. Also like was said, I don't know the conditions that the Los Angeles Times puts on allowing the puzzle to be distributed here. So it's kinda hard to act with too much freedom with the confidence that there won't be any legal issues in becoming an alternative distribution source.
As for automating the process, the only hitch on my end from doing that is knowing where to download the data from. I presume that a lot of the web applets have XML data files that get downloaded/read to show the puzzle. If I knew how to find the URL to this data, it would probably be trivial for me to automate it from there and get the TXT files that are posted. Though I can say I've learned to be very proficient at assembling those files, so I haven't worried that much about wanting to come up with one.