There are four children and their father doesn't call them and they want him out of the picture and give up his rights as a parent what and how can this be done through our local court system without representation

Did you read this? Who would he sign over his rights to? Generally, it can't be done except for an adoption.

If the father is not exercising his parental rights of visitation and contact, then what do they think terminating rights will do?

Courts are very reluctant to terminate parental rights and will only do so in specific circumstances. I don't see any reason for a court to do so in your situation.