Question: Does the Bible require voting by household?
No. The Bible speaks very little about voting in the church, and does not require a certain method. But this doesn't mean any method is as good as the next. Household membership and voting is a helpful way to operate as a church.
The elders are generally to rule in the church. But when decisions need to be made by the whole church, who should vote? If you let any professing believer vote, you may have 6-10 year olds voting on budget or calling a pastor. Hm. Same problem if you let any baptized person vote. Some churches set an arbitrary age. If you have husbands and wives vote, then if they vote alike you just double the vote count, and if they vote against each other they cancel that family out. Also, we should take the marital union into account, and not just reckon every adult as an atomized individual.
I think it is greater wisdom to deal with families on important issues. On lesser issues it can be helpful to have every individual participate, but the church should also put the household head in the position of gathering input from his family, making a decision, and conveying that to his/her family and the church. In other words, this isn't a "men get to vote; women don't" thing. Some women are heads of households. The rest are represented by their husbands. There are times to work with individuals and times to work with households.