I ask respondents to type a list of names into a multi-item textbox. I want to be able to pipe each name separately into later parts of the survey.

You will have to use javascript to turn each line into an embedded data variable to pipe through to the other parts of the survey.I am not a coder but the logic would basically make the input into an array and then for each line assign it to an embedded variable. Then once it is an embedded variable you can pull it through.