Should you ever take a live vaccine while on a potent, oral, liquid steroid for inflammation of the esophagus? yes or no?

Missing information. The body makes anti-inflammatory steroids continuously and treatments evolved to mimic that effect. Whether or not therapeutic steroids interfere with vaccines depends on the type of steroid ,the dose/day and the weight of the kid. The doc who prescribes this should be able to answer this question since they know the data not provided here.

In general, the wisest approach is not to have a live virus vaccine if you are immunocompromised or immunosuppressed. As mentioned, it is best to have the doctor who knows you best guide the interventions.