I have a Python bot running PRAW for Reddit. It is open source and thus users could schedule this bot to run at any frequency (e.g. using cron). It could run every 10 minutes, or every 6 hours.
I have a specific function (let's call it check_logs) in this bot that should not run every execution of this bot, but rather only once a day. The bot does not have a database.
Is there a way to accomplish this in Python without external databases/files?
Generally speaking, it's better (and easier) to use the external database or file. But, if you absolutely need it you could also:

Modify the script itself, e.g. store the date of the last run in commented out last line of the script. 
Store the date of the last update on the web, for example, in your case it could be a Reddit post or google doc or draft email or a site like Pastebin, etc.
Change the "modified date" of the script itself and use it as a reference.
In a Python bot, how to run a function only once a day?