I would like to know if the following is possible and to see if anyone has a template they might share.

I'd like to ask a survey question such as: "Please enter the names of the people on your team." The answer alternatives would be blank text fields, the number and content of which would vary by respondent.

Could the survey be programmed to then use the responses to that survey question to determine how many subsequent question groups to ask and to use the responses in the question stems? For example, if they answered "Tom," and "Sally" to the first item, the survey should then ask questions, grouped by team member, such as...

"How often do you meet with Tom each week?"
"How would you rate Tom's professionalism from 1 to 10?"
"Would you recommend Tom be promoted?"

"How often do you meet with Sally each week?"
"How would you rate Sally's professionalism from 1 to 10?"
"Would you recommend Sally be promoted?"

To be clear, not every respondent will have the same number of team members, and there is more than one subsequent question to be asked about each person named in the first item.

Thanks!

Not directly (What you are looking for is called "looping" usually).

I would go for a slightly different approach. Ask in a question how many members the team has. Based on this you can show and hide question groups. You need to create as many groups of questions as there could be team members.

Then you need to hide the questions and show them based on the response to the first question. You can use relevance equations or conditions for doing that.