Let us look at the answer options --
Option A - Correct answerUses the idiom -- "so + Adjective + that" -- correctly.
Option B - Incorrect"it [=coffee] is of such addiction" -- does not make any sense.
"it is of such addiction, it has become a critical" -- two complete clauses connected by a comma. Run-on sentence. We need a "that" to connect these two.
Option C - Incorrect"addicting" -- I do not think this is standard usage. The correct usage is "addictive".
This option uses an inverted structure. We can rewrite this as -
"It is so addicting as to become a critical part of breakfast for many people" -- this implies that it is very addicting; in fact so addicting that it can become a critical part of breakfast for many people.
But this does not tell us whether it has indeed become the part of their breakfast.
Contrast this with -- "it is so addictive that it has become a critical part of the breakfast for many people". Here we are certain that coffee has indeed become part of breakfast for many people.
Option D - Incorrect"addiction to coffee" is much better than "coffee's [=its] addiction" (this does not make sense).
"coffee becomes a critical part of breakfast for many people" -- Use of simple present does not make sense. We need a tense that conveys that this trend started in the past and its impact is felt event today. Present perfect "has become" should be used.
Option E - Incorrect"there is so much addiction" -- where? in which? Not clear.