The phrasal verb check in can be used in two ways that are relevant to this situation.
I am waiting to check in at the hotel.
This usage is intransitive, and the subject is the customer.
The staff are checking in the guests.
This usage is transitive: the subject is the hotel staff, and the object is the customers.
In your sentence, you are talking about how slow the hotel staff member is, so you must use the transitive version, because the hotel staff member is the subject. The verb check is in its infinitive form, so the subject is moved to the verb take. Note that the customer is the direct object of check, so you don't need the preposition for.
Why is he/she taking so long to check in the customer?
If you don't know that the hotel staff are the cause of the problem, you could say:
Why is it taking so long to check in the customer?
If you think that the current customer is making a lot of fuss and this is causing the delay, you could use the intransitive form.
Why is he/she [the customer] taking so long to check in?