Why did Mary and Joseph look for an inn? Luke 2:7 says:

7 And she brought forth her firstborn son, and wrapped him in swaddling clothes, and laid him in a manger; because there was no room for them in the inn.

This implies that they first looked for an inn (maybe a guest room). I can understand the significance of the Christ child being born into humble circumstances, but why would an expectant couple seek an inn?
And really, would an inn have given them more privacy (which the account clearly shows is something Joseph wanted for Mary)? Joseph and Mary must have known the prophecy that Jesus would be born in Bethlehem, so I doubt the birth was a surprise while they were there.
The only other instance of inn in the KJV New Testament is Luke 10:34, the parable of the good Samaritan:

34 And went to him, and bound up his wounds, pouring in oil and wine, and set him on his own beast, and brought him to an inn, and took care of him.
35 And on the morrow when he departed, he took out two pence, and gave them to the host, and said unto him, Take care of him; and whatsoever thou spendest more, when I come again, I will repay thee.

In this case, the host provided care to the person (in return for payment). If innkeepers sometimes provided care to their guests, was it common for women to give birth in inns?
Perhaps, it seems that you might think there were other options available to them. There simply were not.

In those times, and until very recently actually, women birthed their children at home (or where ever they were comfortable (ie. by the river, a relative's home, etc.)). Also, the idea and treatment of pregnancy and birthing as a medical condition is very modern and very western.

Women also didn't really have the choice to go to the hospital. Most cities worldwide did not even have them! And Bethlehem was a little backyard town.

So Mary and Joseph had only a few options really:


Relative's or friend's home
Inn
Wilderness


The first two were apparently unavailable and the third seems less comfortable and safe than the manger.



Source:


The History of Hospitals
"The shift to hospital births started in the 20th century."
"At the time, Bethlehem’s population was between five and six hundred people." (Not a bad size for an ancient town, but certainly not big enough to spend the resources for a hospital)
AiG makes the claim that Jesus was born in a peasant's home, which looked a lot like today's barns.