Update: other than the built-in codes 001 through 008, which take you to various parts of the MPC website, it looks like:
013 takes you directly to the Camp Moonforest listing
020 takes you to The Lady Afterwards
021 displays a message: "it appears the intern in Room 021 has begun exhibiting signs of extreme fear"
023 takes to to the Doomensions page
033 takes you to the main MPC homepage
042 displayed a message about "protocol 051 initializing", then I wasn't able to get it to pop up again
047 displays a message: "much obliged - unfortunately we are not looking for those services at the moment"
051 takes you to the Disappearance of Cadet Turner page
069 takes you to an erotica ebook about dinosaurs (editor's note: how far have we fallen?)
117 takes you to the Post Mortem page
221 displays a message: "elementary, my dear"
231 displays a message: "k636xwj8f3vhqbgh"
300 displays a message: "yptjjtcy3jk6rfq8...something worth fighting for. First in...best dressed"
314 displays a message: "this is a pie joke"
316 takes you to the Hastur page
333 takes you to the Curious Correspondence homepage
368 displays a message: "it is a giftcard you goose"
404 displays a message: "whatever you're looking for, it's not here"
420 takes you to a Snake game
459 displays a message: "459 | 514 | 569 | 624 | ___" (editor's note: my 6th grade math class leads me to 679)
559 takes you to the blind box product page
664 displays a message: "xm987c7gqxtvtm6r"
666 takes you to the countdown clock
679 displays a message: "Satisfactory. Report back here on December 1st - Curator"
769 displays a message: "it is a giftcard you goose"
801 displays a message: "do not save what is left after spending...but spend what is left after saving. cjdhrhcyb76k9bmy"
888 displays a message: "if I don't have my memories did I ever exist?"
911 takes you to the Post Mortem page
925 takes you to the page where you sign up to become an affiliate
998 displays a message: "f7pxxvy8qc7v8gg3"
999 takes you to the Crypt
There are also buttons for * and #, but those would be an extra ~680 something entries, and my fingers are very tired
I tried googling the alphanumeric responses, but got no results. I also tried doing mysteriouspackage.com/gibberish, but got nothing. So either they're gibberish responses, or they're encoded somehow. For now, it looks like we just head back to the Console on Thursday? And then something launches in mid to late March?