To me, that sentence sounds clunky whether you add an ‘and’ or not.

I would say one of these:

The husband, wife, and mother are coming, as well as the sister.
Both the husband, wife, mother, and sister are coming.

To me, that sentence sounds clunky whether you add an ‘and’ or not.

I would say one of these:

The husband, wife, and mother are coming, as well as the sister.
The husband, wife, mother, and sister are all coming.