Perhaps they did, but nothing in the bible indicates that they stayed at an Inn. 

In fact, it's unlikely, because Joseph had relatives in Bethlehem. 

 

The Innkeeper plays an important role in most christmas plays and stories, although he isn't mentioned in the Bible at all. Many christmas stories are merely myths.

 

''Joseph also went up from Galilee, out of the city of Nazareth into Judea, to the city of Davd, which is called Bethlehem, because he was of the house and lineage of David, to be registered with Mary, his betrothed wife, who was with Child. So it was, that while they were there, the days were completed for her to be delivered. And she brought forth her firstborn Son, and wrapped Him in swaddling cloths, and laid Him in a manger, because there was no room for them in the Inn (kataluma).'' - Luke 2:4-7

 

1. There was NO Innkeeper.

2. There was NO Inn. (as you'll see)

3. Joseph and Mary were NOT in a hurry to find a place to stay.

 

The Innkeeper is simply not mentioned in the Bible, nor anywhere else. 

The Inn on the other hand is mentioned in many of our Bible translations. In fact, the ancient greek word 'kataluma' doesn't mean what we think it does. In modern english we translate it 'Inn' as in hotel, but what it actually means is 'guest room'. And looking at the jewish traditions that time, it's very unlikely that they didn't choose to stay with relatives, especially for Mary was pregnant.

 

NOTICE: The Bible metions the word 'Inn' or 'kataluma' several times. (Luke 22:11 & Mark 14:14)

 

Consistent with the true meaning of 'Inn', the recently updated NIV Bible translates it, ''because there was no guest room available for them.''

 

So rather than desperately looking for a place to stay and constantly facing closed doors, they stayed at their relative's house with many other family members. The registration in Bethlehem must cause an entire family reunion, which explains why no guest room was available. 

 

The Bible also indicates that they weren't in a hurry at all, because Mary didn't give birth to Jesus that same night. (''So it was, that while they were there, the days were completed for her to be delivered.'' - Luke 2:6) 

God's word clearly states, that while they were in Bethlehem, the days were accomplished. It would have also made more sense to arrive in town well before her due date, because such a long journey could have been very dangerous.