I get to work in several first grade classrooms as a literacy coach, and will share with you a couple of the ways I’ve seen this work.
One way–the class has 4 different colors of tubs for their book boxes. Each color group book-shops on a different day. (For example red on Monday, yellow on Tuesday, etc). On their shopping day, the kids in that group get to shop during one round for 5 books from the classroom room library to add to their books from the teacher. (They return as many as they choose.)
Another way–Friday is “shopping day” in this classroom. During each of 3 rounds, a third of the class shops for books to add to their book boxes. It’s a bit noisier that the first example, but works for this teacher and class :).

Hi!
I am always of the mindset that my most important job in the classroom is teaching. So putting books back into their book tubs is not an activity I ever do!
So what is my procedure? Kids do it! SREA have one idea for book shopping. Here is a great article for one of our favorite ways to book shop: https://www.thedailycafe.com/articles/3-ideas-for-book-shopping-during-daily-5?preview=1&version=1
When it is time to shop for new books, students return the books to the book boxes. That said, I also have taught K-6 and know that at every single grade there are some students who are not the best at returning to the correct box. To avoid me ever having to manage the books, each student in my classroom is in charge of one or two book book tubs. 1 day each week I have students go through their tub to check the number on the back of each book and be sure it coincides with the tub number, located on the front of each tub. If the books are in the wrong place, they return them to the correct place.
This can be done as a whole group or have certain students do it on specific days.

I use the following procedures for borrowing books in the classroom:

1. I allow the students to keep one book from the classroom library in their desks (they may keep a personal book as well) for those down times when they need to use time wisely (e.g. I am speaking to someone at the door, etc.). They are allowed to switch these books during recess breaks, free time, etc.
2. They borrow 5-8 books from the classroom library for their Daily 5 book bins. Again, they are allowed to “shop” for new books during recesses.
3. I allow my students to take one book home at a time. Like the libraries I knew of as a student, I put a pocket with a library card in the back of the book, and I model my expectations for signing out the card. Cards go into a container, and are retrieved when a book is returned. I have two daily helpers called “librarians” who not only ensure that books are put back neatly and correctly into their corresponding tubs, but peruse the cards to see if anyone has a book out for too long. They remind that student to bring back the as soon as possible. Primary aged students are more than capable of doing these tasks, as long as it is modelled for them.
   Hope this helps! Good luck!