There are two kinds of logic, deductive and inductive. Deduction is largely tautological: you can manipulate existing data but you cannot discover new truths (as in Socratic syllogisms). Looking at reality to determine an "is" is induction.
Furthermore, the choice to live is NOT an axiom of the Objectivist ethics; there are no ethical axioms, since ethics is a derived branch of philosophy. Ethics is a guide to living. An ethics that did not, in fact, guide you in some method for living would be a contradiction of terms. This is why ethics only applies to the living, and since you have volition, the fact of your continued existence is open to your choice. If it were not, you would not require a guide in order to live (you would do so automatically), and ethics would, again, not apply to you.
This is why Ayn Rand said that nature presents us with a great many "oughts", all of them conditional: IF you want to live, then you OUGHT to eat. If you want to eat, you ought to work. And so on. However, "want to live" is not the "is" here, it is the condition. The fact that you have to eat in order to live is the pertinant fact of reality (the "is"); it is not subject to any of your whims, feelings, etc. It simply is. No matter what you do, you will not be able to alter the fact that your life requires sustenance. However, you can alter whether you continue in existence or not by whether you behave according to the demands of this fact.
That is how you derive ought from is. Those who call this a dichotomy of some kind are, in effect, saying that men don't actually require food in order to live; men can live in any manner whatsoever and reality is malleable to anyone's whim.