"Look, Grandmama. I'm Joseph taking care of Baby Jesus!"
God doesn't leave anything to chance. So when he chose Mary to be the mother of his son, he was also choosing Joseph, her betrothed, as the right man to be Jesus's earthly father.
Joseph was a just man, who wanted to protect Mary from the public shame of being pregnant before she was married. He was obedient to God, apparently without question, even though it would mean sharing her embarrassment (Matthew 1:19-25).
Knowing that Joseph was that sort of man, I wouldn't be surprised if he did actually crawl into the manger to care for their baby.