These
are standard units and have been for many decades. The cap(s) can be just about anywhere; since their location is not important to their functioning (and they outlast most other components) you often find them tucked away in the oddest corners. Capacitors are sold in standard values and you will have no trouble finding what you need at any repair house or through a catalog. The weather proof part doesn't mean they are made to run under water. Do not believe the clerk who tells you you have to adjust capacitance to reflect the higher voltage in use today, blah blah blah, etc, etc. Stick with what is marked on the schematic or the capacitor or stamped into the motor frame. Oh, usually the capacitor will be rated at double or quadruple voltage, that is not important - just not under-rated.
One sure way to find it (them, remember, if this one is shot the others are not far behind) is to trace back the wires from this motor. Of course, we still are not positive that it HAS a capacitor hooked up in series. There are other solutions available here so you may not be out of the woods yet. Oh, one thing which occurs to me: Could it be that this motor is not getting the voltage it needs? You said it is 115v. Does it have two or three wires running to it?
If only two, then try hooking it up to the line directly. If it purrs along, well then, it is something else. It there are three wires, then you probably do have a capacitor in series with the starting winding. Or, other possibility, this motor is not original and somebody made a mistake in hooking it up.
I have kept quiet on the lube front, but if you don't mind my .02cents...I think you should not be spraying WD40 in the housing of this airconditioner. It stinks forever and since this thing constantly moves air into and out of the room, you could well end up with long-term stink.
WD-40 is a good "un-freezer" and a lousy lubricant. In my opinion (ok, more than just my .02cents, sorry) it is best to do the job right: Carefully disassemble the motor, clean the shafts (no sandpaper), re-oil the bearings with a decent oil (20w is cool, if it is non-detergent and non-acid) then re-assemble being careful to tighten evenly and not too tight (hand tight not arm tight) with lock-tite or somesuch.
Such a pretty unit, I bet you will soon have it up and running.