Lymph nodes under the arm, I want to know can they become inflamed from using the monkey bars?

Usually not. Lymph nodes respond to some inflammation or infection, which monkey bars should not cause. Hanging from the hands or other parts of the body should not bother the armpits. However, hitting the bars with the armpits or holding on to a bar with the armpits would squish lymph nodes and probably worsen them. If nodes enlarge for "no reason", a primary care doctor can investigate.