Is "do so so that" in the following sentence correct?

The double so in do so, so that is not a problem.

But you don't have an antecedent for your first so, which is a pronoun that means such as has been specified or suggested. You haven't specified anything. Also change could to can:

Being hospitable is one of the traits of a friendly and welcoming person. And a good host's business is to do so, so that their guests can feel comfortable and make themselves at home.

If you don't like the double so, you can change the pronoun so to that (and drop the following that):

Being hospitable is one of the traits of a friendly and welcoming person. And a good host's business is to do that, so their guests can feel comfortable and make themselves at home.

I would further suggest that you drop their, since you technically don't have an antecedent for it. It's not the noun business. It wants to be the noun host. But you have only host's — a possessive, not a noun. Even if you accept a possessive antecedent (as many do), you still have a mismatch — a singular host and a plural their:

Being hospitable is one of the traits of a friendly and welcoming person. And a good host's business is to do that, so guests can feel comfortable and make themselves at home.

Lastly, here's what I would do:

Hospitality is one of the traits of a friendly and welcoming person. And it is a good host's business to be hospitable, ensuring that guests are comfortable and feel at home.