Can the chickens vent get blocked with eggs?

Yes.It is called egg binding and it is the inability of the hen to pass an egg. The partially developed egg could have either a hard or soft shell, or no shell at all. This inability to pass the egg often results in the death of the hen.

There are ways to help the hen.

Separation from the flock to a quiet area where the bird can concentrate on the job at hand.

Extra calcium in her diet will not affect the offending egg but will help build muscle strength needed to expel the egg.

A water soluble calcium supplement is available.

A small amount of vegetable oil spread around the vent and just inside will help soften the mucus membranes and lubricate.