My daughter is still in high school, but has a full-time college schedule with Running Start. She doesn't pay tuition, but can I claim her student fees and book costs?
Fees that are "required for attendance" qualify. Only books that are required for the course and bought from the school qualify for the Lifetime Learning Credit.
Expenses MAY qualify for the American Opportunity Credit (AOC). See details below. But you may not want to use it. Claiming the AOC is limited to 4 times. Most parents want to save those four times for when they paying more (so getting a bigger credit) later during actual college years.
High school students are not usually eligible for the more generous American Opportunity Credit, unless they are officially enrolled as a degree candidate. From pub 970: "Example 3. During the 2019 fall semester, Larry was a high school student who took classes on a half-time basis at College X. Larry wasn't enrolled as part of a degree program at College X because College X only admits students to a degree program if they have a high school diploma or equivalent. Because Larry wasn't enrolled in a degree program at College X during 2019, Larry wasn't an eligible student for tax year 2019." Reference: https://www.irs.gov/publications/p970/ch02.html