する verb
せんきょする
Tofuguの会長を毎年選挙しています。
We elect a Tofugu Chairman every year.
Election is 選挙 so when you make that word into a verb, you have to elect.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well. Also, the reading just comes from 選挙 so if you know that you know this as well.