The display is entirely driven by the processor section. If the digital side is malfunctioning, you will get no display (or maybe just a dot). And while (obviously), 'no display' can be caused by a problem in the CRT driver cirucits or PSU, most of the time it is a logic problem.
From the Australian museum site you can get the official HP service manual ('boardswapper guide' :-)), the interfacing manual (you may wonder why you need this -- it's actually the most technical of all the 9100 manuals and is well worth reading), but no full schematics. I don't own a 9100A, you see... But there are 9100B scehamtics that might be a starting point.
The 9100A and 9100B are similar in some areas and quite different in others. From what I remember :
The flip-flop boards are the same
The control logic (clock and core-on-a-rope microcode) board is similar, different ROM programming, but otherwise much the same
The core memory system is very different (but similar is design concepts?)
The main ROM/gate board assembly is different (again similar concepts?)
The keyboard is pretty similar.
The card reader is different
The entire top case, PSU and CRT circuitry is the same.
Now, getting started...
Firstly, I would take the whole machine apart (this includes removing the 'sideboards' that connect the ROM to the gating board) and clean all the edge contacts. Look for silly faults too at the same time.
Check the 3 fuses (mains, -15V supply, EHT supply) on the PSU chassis in the top case. Also check the 3 little filament lamps up there, you need them for the tests.
Reassemble, but leave the card reader out for the moment. You must connect the keyboard (the machine will not run if there keyboard is not present). Power up.
Check the -15V supply to the logic. Normally, if the 2 lamps that illuminate the register label to the right of the CRT are lit then the -15V supply is OK, but it's worth checking.
Press STOP twice (in case the thing is stuck in a running program), the do something that will cause an error -- 1, CHS, SQRT is my normal sequence. If the error lamp (left of the CRT) comes on, then the processor is basically working, so troubleshoot the CRT driver circuits
Check the CPU clock is running (test connector on the control logic board, leftmost in the cardcage). Check the various bias supplies on that connector (and on the flip-flop board connectors).
Now, some of the flip-flops are the program counters for the microcode and firmware ROMs. I can look up which ones they are for the 9100B, but the -A may well be different. If you can find them (I can give you hints...) then check the outputs (E and F signals) on the test connectors on the flip-flop boards. All should be toggling. If not, either the flip-flop is defective, or the logic driving it, or...
A 'silly' that will remove the display is a lost of the mains sync signal (the display routine is synchronised to the mains to prevent flicker). In the 9100B it's a transsitor on the right sideboard -- no idea where it is on the 9100A, but tracing the AC signal in the PSU cableform should find it.