Do two subjects need to be connected by a conjunction

Something has gone wrong with the sentence you quote, for sure, but adding a conjunction before "requests" doesn't really help. I'd suggest replacing that last observation, beginning with "requests", with

(Such requests for highly paid professionals have increased in the recent wave of corporate takeovers.)

I can parse the original example in this way: "requests for whom have increased in the recent wave of corporate takeover" is an extraposed relative clause which modifies "highly paid professionals". Before extraposition, this has the form

... when highly paid professionals requests for whom have increased in the recent wave of corporate takeover are placed.

which, IMO, is grammatical, though very awkward, because of the overly complex subject. Sometimes, in such constructions, the awkwardness can be avoided by moving ("extraposing") the relative clause to the end of the clause. Doing this leads us to your example sentence. But something goes wrong. I don't know what constraint has been violated, but the result is, IMO, ungrammatical in English.

I don't doubt that you've quoted the example correctly, and as I've just said, I don't know why it wouldn't be grammatical. But nonetheless, IMO as a native speaker of English, that sentence is no good. It just isn't grammatical.

Temporary-employment agencies benefit not only from the increasing demand for clerical workers but also from the higher profits made when highly paid professionals are placed, requests for whom have increased in the recent wave of corporate takeover.

If we simplify, we have:

They benefit from this and from profits from some people for whom requests have increased.

There are not “two subjects”: there two prepositional adverbial clauses

(i) from the increasing demand for clerical workers

and

(ii) from the higher profits made when highly paid professionals are placed

The prepositional adverbial clauses are joined by the conjunction “but”