@Chics4me
Hi
I doubt she is egg bound with that amount of swelling but to check you need to lube a gloved finger and insert it into her vent about an inch. If she has an egg stuck, that is most likely where it will be and you should be able to feel it. Usually they will die within a couple of days with being egg bound because it blocks the elimination of waste (pooping) and the bird dies of a build up of toxins in the body.
Sometimes a shell less egg will cause them issues in trying to pass it and the vent will pulse but again it will not cause that amount of swelling.... that said, a dietary imbalance can cause them to lay down significant fatty deposits in this area which can make it look swollen like this and can also cause them to lay shell less or soft shelled eggs, so that might be the issue.
What do you feed them, including treats? It looks like she is a buff Orp and being larger birds they are more prone to developing these thick fatty deposits.
Other possibilities are Ascites, Internal Laying possibly leading to Egg Yolk Peritonitis or Salpingitis.
The fact that her vent is pulsing suggests it is a soft shelled egg or Salpingitis as contractions of the oviduct are probably causing the pulsing. However it is also possible that the pulsating is due to the rear air sacs either side of the vent heaving due to pressure being put upon the respiratory system by a mass of fluid or egg material caused by the other ailments mentioned..... does she appear to have any respiratory distress ie is she open mouth breathing/panting....if so, is the rhythm of this in synch with the pulsing of her vent?
Can you take some video footage of her to help us see what you are seeing? It needs to be uploaded to the likes of You Tube or Vimeo and then post a link to it here.
Does her belly feel like a water balloon?
Is her lower back rounded and hunched?
Has she laid weak shelled eggs recently?
Does she walk with a wide stance like her hips are being pushed apart?.... compare with other chickens to get a better idea of what is normal.
Do You know when she last laid an egg?
Have you seen any lash eggs in recent days/week?
Unfortunately it is not easy to tell the difference between Salpingitis, internal laying/EYP and/or Ascites. The main difference with Ascites and internal laying is that the belly hangs lower between the legs whereas the impaction of the oviduct caused by Salpingitis is usually suspended a little higher in the body.
Sadly with most of these ailments the long term prognosis is poor. A soft shelled egg and or dietary imbalance is probably your best hope. Knowing what you feed them may help to indicate if this could be the issue.
I'm sorry that I am unable to give you clear and concise things to look for to figure this out. Vets sometimes resort to Xrays to try to determine the nature of the problem but even then they can get it wrong, as a recent thread here highlighted.