No. There is no direct connection between Peter Billingsley, who is known for 'A Christmas Story', and Iron Man. They are from different movie universes.
Peter Billingsley's role in 'A Christmas Story' is that of a young boy in a family - centered, Christmas - themed story. Iron Man is a high - tech superhero. They are completely different in nature.
In 'A Christmas Story', Peter Billingsley is famous for bringing the character of Ralphie to life. Ralphie's quest for a Red Ryder BB gun was a central part of the story, and Billingsley's acting made us feel all of Ralphie's emotions - from the excitement of the idea of getting the gun to the fear of what his parents might say. His expressions and delivery of lines were key to the movie's success.
Peter Billingsley was significant as he made the character of Ralphie so endearing. His acting made us care about Ralphie's quest for the BB gun. In the movie, his performance held the whole story together. If not for his great portrayal, the movie might not have been as successful in becoming a Christmas classic.
Since we don't know for sure if he paid for it, it's hard to say the reason. Maybe if he did, it was because he had a strong passion for the project and wanted to ensure its success.