Well, I can tell you that in the hospital, we’re most worried about fluids. Water, because of the impact it has on electrolyte distribution, is a killer both in having too much and too little.
After that, we can look at the electrolytes themselves, which are minerals, not vitamins. Potassium is first on the list. Your body doesn’t make it, nor store much of it. Potassium is needed for muscle relaxation after contraction, including the muscles you need to breathe. Severe hypokalemia (low levels of potassium) with cause respiratory arrest because the muscles won’t relax enough to get a good contraction again.
Sodium’s probably next. Too little sodium and we see brain swelling and neural effects.
(Really, potassium and sodium are equivalently vital. However, true sodium depletion is rare, because the body is good at recycling it instead of excreting it. We often need to supplement potassium in the hospital, rarely do we need to give sodium. If a person’s sodium level looks low, it’s usually because they have too much water in them, not because they have too little sodium.)
After that, I’d venture calcium. Although, as a doctor once told me, “there’s really no such thing as hypocalcemia - you’ve get all the calcium you need in your bones even if you have osteoporosis…the trouble is getting it out of the bones and into the blood where it’s useful.” But functional hypocalcemia will also cause problems with cardiac and skeletal muscle contractions.
The most important vitamin I can think of is Vitamin C, required for wound healing and skin integrity. If your skin breaks down, you invite infection, and that will kill you quicker than other vitamin deficiencies.
A person who can’t eat or digest things through their gut will get water and glucose (carbs) through an IV if we think they’ll be better within a week or so. Some docs like to add a vitamin blend at this point, but most hold off. If their electrolyte levels get low, those are added next. A person on full parenteral nutrition will get amino acids (proteins) and lipids (fats), as well, but it can be some time (days or weeks) before we decide to put them on those. So water, glucose and electrolytes are more vital than proteins and fats, and docs disagree on when vitamins are needed.