My 300SD just got fixed for a similar situation.
After locking drivers door, electric pump in trunk comes on,
other three doors struggled to lock.
I experimented with quickly (while pump was running) pushing
down or pulling up on the other three door locks.
They moved very easily, which told me vacuum was present.

When I manually pulled or pushed the passenger door the two rear immediately
followed up or down.
Only the drivers door is mechanical, the other three are vacuum.
Admiral: my 300sd has single vacuum port on door lock actuator,
the 80 and 81 300D have dual port actuator.

What to do: Admiral is right that it is a door actuator. Take the door panels
off, unplug vacuum line from actuator (you can do it without unbolting anything - except the door panel). I plugged mine with golf tees while testing.
If you isolate the bad unit, the rest of the system should work.
After finding the bad one, it is easy to drop it out of the door and replace.
Will not cost very much for a new one (from Fastlane).

In my case that passenger door was the only bad one, replaced it and
I have working locks again. In my 300D it was the trunk actuator.
Replaced it, and all my locks work again.

One more note: I used a 300D dual port actuator to replace my 300SD single port
door actuator. You can search for this W123 into W126 post.
Have to use the correct port to push/pull and it worked.

__________________
80 300D 340K Owned 30 yrs
83 300SD 440K Owned 9 yrs - Daily Driver 150mi/day
02 Z71 Suburban 117,000
15 Toyota Prius 2600 miles
00 Harley Sportster 24k
09 Yamaha R6
03 Ninja 250