For a fall 5th completion you definitely don't need a sandfish in the cart, just fish one up once you open the desert. The vault bundle is available once you've completed 4 other bundles which can be done pretty easily in Spring (spring Forage and spring crops are easy, and they unlock the boiler room which can be fully completed once you get to level 80 in the mines. Crab pot and Construction bundles can also be completed in spring).
You should also be able to upgrade your barn and raise a pig in time to get a truffle by fall 5, so although one appearing in the cart would be useful it isn't essential. A Rabbit's foot might drop from rabbits in time but I usually get my first one from serpents in skull caverns.
If you don't mind resetting days, you can get a Nautilus shell in the mail from Demetrius. Gift him quality Strawberries to raise friendship and when he sends you a letter restart the day until you get a Nautilus shell. They can also be produced by fish ponds but that's harder to guarantee. Pre 1.6 you could use Blade's map predictor tool to tell you where to put the pond to guarantee Nautilus shell but I don't think that's an option right now.
What you will need in order to complete the whole thing by fall 5 is a crocus and a snow yam OR eggplant, red cabbage and pumpkin all appearing in the cart in time to unlock the greenhouse and grow a batch of winter seeds by the 5th. (To get the winter seeds, put winter roots from frost slimes in a seed maker or trade spring seeds with the desert trader on Saturday. Speed up the growth with deluxe speed growth bought from Sandy on Thursday if necessary.)
