Do ebooks purchased for an online class and lab equipment count for "Books & materials required to be purchased from the school" if they were purchased thru a 3rd party?

Yes, it doesn't matter how you purchased the books needed for your education. Please see below on qualified education expenses:

The expenses were not available for purchase directly from the university, they were required to be purchased from a single 3rd party source. Do they still apply to the  Lifetime Learning Credit?

Yes if this source was the way the institution instructed you to buy the books.