Making Eggs for 50 people

I highly recommend using a steamer if you have access to one, but if you don't have it, you don't have it. The good news is that the process for making scrambled eggs on the stove is about the same no matter how many eggs you're making - add as much beaten eggs as you're comfortable stirring in the pan, cook over low heat stirring occasionally.

The major tips/gotchas that I can think of:

1. Open all your egg cartons at once before you start cooking. The last thing you want to be doing is fighting with packaging material when you need eggs quickly
2. Low heat, be patient, stir frequently but not constantly. Once every couple minutes should be fine depending on the size of your pot - it shouldn't be hard to figure out if you need to be stirring more or less frequently. However, don't forget to stir. This is probably the biggest way for things to come out less than perfectly. If you get distracted and forget to stir the eggs for too long, the ones at the bottom of the pot can start to develop the same green off-color as overcooked hard boiled egg yolks. It's perfectly safe to eat but who wants to eat green eggs?
3. You mention butter but unless you're adding it for flavor it's honestly not necessary. When you're done there's going to be eggs stuck to the pan no matter what you do. Don't stress yourself out about it.
4. One tip - if your eggs are cooked more than you want them to be but not at the turning green stage, you have some wiggle room in whisking in uncooked beaten eggs into the overcooked eggs. The heat from the cooked eggs and further hot holding will cook the raw eggs and it will distract somewhat from the overcooked bits. It's not perfect but it's "good enough" and it can help you save a chafing dish of eggs that you'd otherwise have to throw out.

That's really all there is too it. If you have more eggs than you can comfortable fit in the pot, you'll have to do it in batches but all that means is "cook eggs, dump into chafing dish/hotel pan, start cooking more eggs."

As MikeTheLiar mentioned, a steamer would help for this:

If you have more than one pan to cook them in of the same width & length by different depths, and they're not thin disposable containers, you might try making your own bain marie (double boiler) :

Place the larger one across the burner or two, and put an inch or so of water in it. (you should try for an inch (2.5cm), but you might need less if the depth of the containers is too similar, and placing the top container on forces water out of the lower container).

Put the other container on top.

Once the water is boiling, and you have steam, adjust the temperature so the water is staying hot and generating steam, but not a rolling boil.

Add your oil or butter, let it coat the pan well, then cook the eggs in the top vessel, stirring constantly 'til it's not quite to the consistency you want (as there will be a little bit of carry over-cooking)

...

I've also heard that you can do large batches of scrambled eggs in the oven (350°F/175°C), stir after 10 minutes, then every 5 minutes 'til they're done (technically, not quite done) ... most recipes claim it takes 20-25 minutes.

From personal experience, I have baked scrambled eggs for large numbers of people (100+ in some instances). I used a recipe similar to this, literally the first link that came up when googled. I have used steamers in the past but found the oven easier for consistency sake, without having to seek out new equipment.