Is it legal for a passenger to sit on the jump seat in Part 135 operations?
Lets start with 14 CFR 135.125, which defines security requirements for part 135 flights:

§135.125   Aircraft security.
Certificate holders conducting operators conducting operations under this part must comply with the applicable security requirements in 49 CFR chapter XII.

We pass the ball to 49 CFR 1544 and make sure we're in the right place:

§1544.1   Applicability of this part.
(a) This part prescribes aviation security rules governing the following:
(1) The operations of aircraft operators holding operating certificates under 14 CFR part 119 for scheduled passenger operations, public charter passenger operations, private charter passenger operations; the operations of aircraft operators holding operating certificates under 14 CFR part 119 operating aircraft with a maximum certificated takeoff weight of 12,500 pounds or more; and other aircraft operators adopting and obtaining approval of an aircraft operator security program.
(b) As used in this part, “aircraft operator” means an aircraft operator subject to this part as described in §1544.101.

For brevity, I'm not posting 1544.101 which defines a host of paragraphs you must comply with depending on your operation.  If we assume a charter operation in an airplane weighing over 12,500 lbs, we get (1544.101(e)(1)):

(1) The requirements of §§1544.215, 1544.217, 1544.219, 1544.223, 1544.230, 1544.235, 1544.237, 1544.301(a) and (b), 1544.303, and 1544.305; and in addition, for all-cargo operations of §§1544.202, 1544.205(a), (b), (d), and (f).
(2) Other provisions of subparts C, D, and E that TSA has approved upon request.
(3) The remaining requirements of subparts C, D, and E when TSA notifies the aircraft operator in writing that a security threat exists concerning that operation.

From here the one we care about is 1544.237, which says:

§1544.237   Flight deck privileges.
(a) For each aircraft that has a door to the flight deck, each aircraft operator must restrict access to the flight deck as provided in its security program.
(b) This section does not restrict access for an FAA air carrier inspector, an authorized representative of the National Transportation Safety Board, or for an Agent of the United States Secret Service, under 14 CFR parts 121, 125, or 135. This section does not restrict access for a Federal Air Marshal under this part.

Which brings us to our answer: YES, a passenger may occupy the jumpseat if you do not have a door, and also if you do have a door as long as your company security program allows it.