Plant grape instead.
Although technically "toxic" the chickens would most likely sample the morning glory and move on without ingesting nearly enough to be of concern. However, when confined to a run, it may be the only greens they have access to and will pay more attention to it than they should.
Plant the grape outside the run and allow it to climb the fence. It will provide shade and food but the vine and roots will be protected from the birds.
Toxic plants need to be researched to see how much is necessary to cause harm. Most of the time its a very large amount that no chicken would ever eat unless it was their only food. Chickens are usually very good about avoiding the bad stuff especially when they have high quality feed available.
My birds free range through our woods and fields and have numerous "toxic" plants available to them. They pay no attention to them.
