It's really difficult to answer, as there are way too many variables.
- Which vegetable?
- What is being done to it before storage?
- What will be done to it after storage?
- Is it closer to 3 days, or 5 days?
For instance, let's look at carrots, one of the items that you mentioned. Cut carrots can be stored in water, in the fridge, and they'll do just fine ... but if you're going to try stir-frying them, the extra water has to be dealt with, and so that won't work (unless you remember to drain them and give them time to dry before you're going to use them.)
Brocolli's another strange one ... if you're slicing up the stalks for using later, once again, water's a good idea ... but I've never had good luck with soaking the florets, as they just take way too long to try to dry back out to use.
Potatoes are one of those things that you might want to do more than just cutting or peeling them -- I typically bake a few off to use in later dishes, like hash. (and I leave 'em whole, and cut them up as I need them). But if you're going to leave them raw, you really should put them in water, or they'll oxidize and turn brown.
The time's also another issue -- not everything can be stored for 5 days without freezing, but that brings in lots of other issues. If I blanch brocolli, 3-5 days is what StillTasty gives in the fridge. (they claim the same for raw, but I know I can keep brocolli from the farmer's market for 7 days ... part of it's buying firm heads .. if the florets give at all in the store, you're not going to get very long out of 'em).
... if it were me, I'd also try to look into what issues your friend has ... there might be tools that could help her (eg, if it's an issue that she's unsteady with a knife, could she use a box grater, food processor or mandoline for slicing things?). If it's an issue with grip, would a vegetable peeler with a larger handle give her more control?