Just sharing my two cents here. It's a good question mixed with modifier and parallelism errors
In 1984 medical researchers at Harvard and Stanford universities concluded that sedentary life-styles lead to heart and lung diseases that shorten lives, strongly recommending middle-aged people to undertake some form of regular exercise.
A. strongly recommending middle-aged people to - here it seems like "recommending" (verb-ing modifier ) is modifying "sedentary life style" which is not the intended meaning. Hence, incorrect
B. strongly recommending that middle-aged people should - If you look closely "recommend" is a subjunctive verb and whenever you use subjunctive verb, avoid using "should" along with it. Grammatically Incorrect, hence out
C. and strongly recommended for middle-aged people to - apart from idiom error i see a parallelism error here, to make it sound grammatically correct it should be Universities concluded that bla bla and strongly recommended that (use of subjunctive verb is appropriate here). Hence, incorrect
D. and their strong recommendation was for middle-aged people to - same as C
E. and they strongly recommended that middle-aged people - Voila! we have found the correct answer. Here, they refer to researchers and is free of errors.