Option B seems to be the best answer.
Option A -
"which" refers to Italy. Incorrect.
Also, option A is a run-on sentence. Two independent clauses are connected by a comma.
"... the study of medieval archeology has prospered ..., numerous towns and regions up and down the peninsula have raised ..."
Option B -
"as" seems to be used in the same sense as "because".
the phrase "backed by a campaign led by the journal Archeologia Medievale" is a participial modifier that refers to the nearest noun phrase "numerous towns and regions up and down the peninsula".
Option C -
Usage of "in that" is incorrect. in that = in this sense.
"in that" is used to describe a "symptom", not a "cause"
for example -
the new computer is better in that it has a faster processor. (It is better in this sense -> it has a faster processor - this is a symptom of "better")
Here, it says that the study of medieval archeology has prospered in this sense - "numerous towns and regions have raised funds". (does not make sense).
"numerous towns and regions have raised funds ..." is a cause (not a symptom) for "has prospered". We need "because", not "in that".
Option D -
Also, option A is a run-on sentence. Two independent clauses are connected by a comma.
"... the study of medieval archeology has prospered ..., numerous towns and regions up and down the peninsula have raised ..."
Option E -
Run-on sentence like other answer options.
Usage of "due to" is incorrect. "Italy" was not due to (= caused by) a campaign.
Also, note that "due to" is a preposition and must be followed by a noun. "backed by ..." is not a noun.