This might be the wrong place to post an answer (hypotheticals when writing are best posted on Worldbuilding.SE) but you don't seem to have a strong understanding of how "pills" (specifically pills given for a medical condition) work.
Most medications for long-term medical problems take a few days to reach steady state, that is, a concentration in the blood where the amount of drug in circulation reaches the therapeutic range and stays there as long as the drug is taken. That essentially means that the amount taken in/absorbed is equal to the amount lost through metabolism and excretion. It usually takes about 5 doses of the drug to reach steady state; if the drug needs to reach steady state more quickly, a "loading dose" of the drug is given.
The same is true of how quickly the drug falls out of the therapeutic range once it stops being taken. The drug isn't instantly out of your body when you miss a dose; it takes a while before it's subtherapeutic, and longer before it's eliminated completely.
So it's unlikely that missing a dose would result in the illness suddenly manifesting itself after one missed dose. The "patient" should have enough time to get a prescription refill before something bad happens; no need for an IV injection of the drug.
Most drugs, even in hospitalized patients, are given p.o. Where an immediate effect in needed (e.g. for pain control, to treat a serious infection, etc.) the drug is given IV initially. Besides drugs for pain, infection, serious allergic reactions, dangerously high blood pressure, fluid buildup in the lungs, cardiac dysrythmias, anesthesiology and a few other conditions/circumstances, I can't think of many drugs that would need to be given intravenously to be life-saving.
What is “Steady State” and why is it important?