The Engineering department is one of the newest departments at the

My answer is (B). It took me 55 seconds.

(A) "than every department at the university" is illogical. Use "than every other department at the university" instead.
"its graduates have high demand" is not sensical as the clause means that its graduates have ownership of high demand.

(C) "they" has no antecedent.
"than every department" is illogical.
"than anyone else" may be interpreted to mean "than anyone else is demanding their graduates".

The use of "yet" is okay.

(D) The first "than every department" is illogical.
The second "than every department" does not make sense.

(E) "they" has no logical antecedent. The only plural noun is "its graduates".
"are demanded highly" is unidiomatic.

The Engineering department is one of the newest departments at the university, but because tits graduates have high demand in industry, it has more extensive facilities than every department at the university.

A) but because its graduates have high demand in industry, it has more extensive facilities than every department -> Every department or every other department, here 2nd one is better. Another error, "graduates have high demand" doesn't make sense. Incorrect.

B) but because its graduates are in high demand in industry, it has more extensive facilities than every other department -> It is better. Both the errors identified are corrected in this option. Let's keep it.

C) yet they have more extensive facilities than every department because industry is demanding their graduates more highly than anyone else -> We have singular subject, The Engineering department, so there is no need for plural pronoun, "they". Incorrect.

D) yet it has more extensive facilities than every department because industry has been demanding its graduates more highly than every department -> It has similar errors as A does. Incorrect.

E) but because its graduates are demanded highly in industry, they have more extensive facilities than every other department -> they doesn't have clear antecedent. Incorrect.

So, I think B.