Having multiple similar fields is not normalized structure. It can complicate some search efforts. If you want to retrieve all records where Contact A is Emergency contact, must apply filter criteria to all the Emergency fields. And if you wanted a report that listed records by Contact, how could the records be grouped to show all Contact A records together? Now would need to use a UNION query. Also, as you noted, if you want to add another Emergency field, need to modify design of table, queries, forms, reports.
It is a balancing act between normalization and ease of data entry/output. Non-normalized structure looks easy for data entry but can complicate some output. Can be handled but do you want to?