Substring search on multiple attributes
You should create full text search indexes for these fields and use MATCH ... AGAINST

This way you can change this line to:

$query = "SELECT * FROM coils MATCH ( name, resistance, wraps, wire_1, wire_2, wire_3, wire_4, wire_5, wire_6) AGAINST ('$term') LIMIT 25";


Updated: If you want stick to string substitution when mounting your $query (like you did), make sure you have sanitized your $term variable in order to prevent SQL injection attacks. But it's highly recommended using bind solution nicely expanded below by @greg-burghardt.

More information about MATCH AGAINST can be found here: https://dev.mysql.com/doc/refman/5.7/en/fulltext-natural-language.html