I was thinking more of VC-135A as meaning Vomit Comet 135A
Oh! As a purist, I tend to the original system brought in about 1962. So to me a VC-XXX is a cargo aircraft (C) modified to carry VIPs (VC). Like the VC-137. Let's not discuss the KC-10 or KC-767.
Some people are not as familiar with the system, so perhaps I should expand further.
WARNING! SPOTTER TALK AHEAD!
This is the Military Designation System (MDS, sometimes referred to as the Mission Design Series) to which I am referring:
https://en.wikipedia.org/wiki/1962_United_States_Tri-Service_aircraft_designation_systemIt is the common aircraft designation system for all the US military forces since the early 1960s.
The C-135 is the military equivalent of the Boeing 707. In fact, when Boeing first field the concept, it was at the company's own expense and a two pronged approach: a new jetliner to the commercial operators, and the KC-135 Stratotanker to replace the existing fleet of KC-97 piston-driven tanker aircraft (In fact the tanker was the primary drive with the jetliner a nice bonus) which had increasing problems flying fast enough for the jet fighters & bombers to safely use. They also foresaw that an airliner could also be used as a cargo aircraft, and the USAF had a growing need for a jet powered cargo aircraft. This is why I get a little miffed when people say that the C-135 is a modified 707; it isn't. The C-135 and the 707 are BOTH variants of the original Boeing Model 367-80 (often called the "Dash 80") technology demonstrator aircraft.
A little surprisingly, the tanker came first. Under the MDS, this perhaps being a tanker should have been called the K-135A ('K' standing for "tanker" and the first letter indicating its Primary mission) but it was called the KC-135A (C - Cargo, primary mission; K - Tanker, modified mission).
Later came the C-135A (pure cargo aircraft) and lots of different varients (EC-135, RC-135, etc).
So we had this tanker aircraft called the Boeing KC-135A. NASA used some of the ex-USAF aircraft for reduced gravity training and research (NASA rarely bought it's own aircraft; it tended to use USAF / USN cast-offs). This started with C-131 propeller driven aircraft for the Mercury astronauts but was later replaced by a modified KC-135A. At first the USAF provided this service but later NASA took over the programme and the aircraft were repainted in NASA colours under NASA ownership (NASA ended up using four of the KC-135As and a single C-135A); the first three under USAF ownership, and the final two in NASA colours).
Putting aside the outlier, we had KC-135A aircraft who could no longer / never had the ability to tank. The KC aircraft did have the boom fitted (sans "flying wing") and I suspect this was for aerodynamic reasons. The boomer station was still there. Again, I suspect it was for some aerodynamic reason and the cost of removing the gear.
(Side note: In the 1970s, the Royal Australian Air Force (RAAF) was buying "fleet" cars for its use. These were normally the cheapest, base model. When they started getting the Chrysler Valiant - or more properly the Ranger, being the base model - a new thing happened: they came standard with AM radios. This was kinda new but the Department of Defence, in its infinite wisdom, decided they didn't need them. Instead, they paid $15 extra per car to have them removed! And in those days, that was a not insignificant sum. BTW, no, they didn't need the space to fit a military radio. Only a few were radio-equipped, such as Air Traffic Control vehicles, and they had radios fitted under the dash. Government logic.)
So I believe that the aircraft should have been re-designated C-135As again (cargo aircraft), or perhaps NKC-135A (N meaning a 'permanent modification for special test' aircraft).