Broken Arm on Eames Lounge Chair  

So. . .
Broken wood ? Or just a loose part ?
If broken, can we see a photo ?

No broken wood: just good ole "chair lop"
No, the wood is completely intact as is the entire chair; the problem is that it is lopsided (one arm only) because some screws must have fallen out which caused something inside the arm itself to "collapse." The entire right side (arm rest?) moves independently and seems have disconnected from the chair itself. If I physically "lift" the arm up and press it against the chair, nobody would suspect there is anything amiss.
I can take a photo and post it. (Not good at fixing anything, but super super good at photographing anything) I hope I am describing this accurately.

Sounds like...
... your shock mount has come loose from the seat shell. Search the forums and you will find threads regarding this repair. Unless there is significant loss to the wood or the rubber shock mount has sheared in half, you should be able to have it simply re-glued to the wood shell.
The entire chair comes apart with a screwdriver, so get to work and see where it has come apart.

You
won't be able to complete your repair if screws or other hardware are missing. So, you'll need to identify what's needed, and acquire it first. Then the reassembly should be relatively straightforward.
Owners who don't have tools, and don't know one end of a screwdriver from the other, are advised to get professional help.

Knows when to hold 'em, knows when to fold 'em (but don't know how to post 'em...)
Indeed this does sound like the case from the descriptions of the two posts above.
No damage to wood, or to arm padding.
Yes, I recognize that the entire chair does come apart with a screwdriver. I have several screwdrivers, different types of heads, which vary in size, even have a variety of magnetic flat heads which fit on my drill.
I tried searching the threads for this repair before posting here (found nothing, but will try again) as so many of the other threads were remarkable in their visual and instructive detail.
Insofar as my "Rate Your Capacity to Fix this Great 20th C. Furniture Piece At Home Alone" is concerned: I know my limitations and where to pull the proverbial plug before the night ends in tears.
I appreciate the help here. I will send a photo.
Actually, I'll send several, but those belong to other threads; ain't like I'm "design stoopid." (Or "stupid" -- whichever. I swear: I can spell.)
Damn, you guys are tough, tough is good.
Chair is in my "professional" office...I'll send a photo, maybe you can help me i.d. exact nature of the fault if I cannot figure it out.

Did not mean to sound "tough"
Just trying to help.
If my guess is correct, you can find out very easily. Just take off the armrests by removing the screws on the underside. When they are out of the way you should be able to see where the chair has come apart. If, as I suspect, the shock mount has released from the shell, I'd quickly unscrew the other side as well so there is no excessive stress put on that connection. I'd leave the top half of the chair off until you are able to have the shock mount reglued or replaced. Then it can all go back together.
Best of luck. I hope it's an easy fix and you can resume enjoying your chair.

I have
not owned or worked on one of these chairs. My literature describes the arm as "twelve-gauge steel plates, bonded with foam rubber and upholstered with matching leather.: No fasteners are visible; does the leather come off to expose bolt-heads on the inside of the arms ?
People ?
(The finish is described as hand-rubbed wax. Is that still the case ? Surprising. . .)

You can see the screws
... but you have to be all the way underneath. They are otherwise concealed from sight. The padded arm comes off and there is another metal plate that holds the seat shell and the lower back shell together.
My understanding is that there was no clear coat on the rosewood shells. Rather it was an oil finish. I think I mentioned this before, but back in the day, Herman Miller would supply the customer with a small can of gunstock oil when they bought a new Eames lounge. You were instructed to oil the shell periodically.
I don't know if they finish the wood on new production.
This illustration might be better than me at explaining the assembly of the chair:

Chair repair
(edited by DA - no advertising please)

One way to approach
a self-repair would be to disassemble the intact side of the chair, and then use the loose fasteners to determine what's missing.
People, do the shock mounts there stay attached to the chair or the arm, when it's apart ?

They are attached to the shells
The front armrest shock mounts are glued to sides of the seat shell,while the rear shock mounts are attached to the backrest on the inside of the curved "lobes".

So. . .
the only small loose parts would be bolts ("screws") . . . ?

Herman Miller Eames Lounge Chair Shock Mount failure
(edited by DA - no advertising please)

Determining whtats wrong
Peter - go away, you're annoying and unhelpful.
First: under the arm you should see two hex-head bolts. An Alan key should allow you to unscrew those. These two bolts hold the upholstered part of the arm to the steel plate underneath. Remove the upholstered arms and set aside.
Then: Inspect the steel plates. They should be connected to seat of the chair by two bolts, and to the backrest of the chair by two bolts. If those are the missing bolts, just screw them back in - but tightly this time! 🙂
If that's not the case: The bolts in the step above are threaded into hard rubber bushings/shock-mounts. These mounts should be glued good and solid to the wood under the steel plate. If one of these bushings has pulled away from the wood you will need to have it re-glued.
I highly recommend (as do most people who have a brain and a sense of resale value) that you use a repair person who will reconnect the rubber bushings. I would highly discourage you from using a service that puts in a non-OEM (original equipment manufacturer) part. Frankensteining a chair like that will kill the resale value.
Personally I have used Alfie Hume on three different chairs and had great results - on one chair I couldn't even remember which mount had come unglued. You could also send the part back to Herman Miller for repair. I'm not sure who has better pricing these days.
Below is the chair, naked of all its padding. Everything should be pretty visible.