What difficulty would arise if a person were born without a thymus gland?

One can live without a thymus gland in a sterile environment. Otherwise one cannot live without a thymus gland since there would be no way one can produce antibodies to fight off bacteria, fungi, viruses, etc. The thymus is the main gland that produces the most antibodies.

You can lead a very normal life without a Thymus gland. If removed in adulthood, other organs (such as bones) take on the production of T-cells for immunity. T-cells have a very long life and can support immunity requirements throughout a normal life span