Updating the item value in the array, and how can I append the item instead of overwriting it?

You have two big culprits in your code. The first one is a flawed control structure:

foreach ($cartItems as &$cartItem) {
    if ($cartItem['bookId'] == $_GET['bookId']) {
        $cartItem['quantityInput'] += 1;
        $found = 1;
    }   
    // If the book is not in the cart item, then append the book.
    elseif ($found == 0) {
        // add a whole new book
    }
}

Since this is inside a loop, the elseif portion will execute each time the id doesn't match, until the book is found. If the book was not present, it would execute for every element of the cart.

That's where the second culprit comes in. You pass the element as a reference in your loop, meaning any operation over that element would change it. Since you were doing exactly that in the elseif branch by assigning it a new value ($cartItem = ...), it would ultimately result in it being overwritten.

What you actually need to do is extract the contents of that branch outside the loop, because you only want it to run once - if the book was not found:

foreach ($cartItems as &$cartItem) {
    // If the book is already in the cart items
    if ($cartItem['bookId'] == $_GET['bookId']) {
        $cartItem['quantityInput'] += 1;
        $found = 1;
        /* Adding a break here would stop the loop once the book was found.
        It would be a logical addition if no special cases exist where
        an item can be added twice independently (seems like that's what
        you're going for, no duplicates in the cart). */
    }
}
// If the book is not in the cart item, then append the book.
if ($found === 0) {
    // everything that was in your elseif branch
}

This should solve your issue. But please consider the following improvements to your code...

This code block is repeated twice (both when the book is not found and when the cart is empty):

while ($book = $result -> fetch_assoc()) {
    $cartItem = array('bookId'=>$book['bookId'], 'title'=>$book['title'], 'price'=>$book['price'], 'quantityInput'=>1);
    $cartItems[$book['bookId']] = $cartItem;
}

The sensible thing would be to extract any duplicating code into a function and just call it multiple times. But in your case this duplication can be eliminated. You need the exact same operation in those two cases and you can have the same code execute it:

if (isset($_GET['bookId'])) {
    $bookId = $_GET['bookId'];
    $found = 0;
    foreach ($_SESSION['cartItems'] ?? [] as $cartItem) {
        if ($cartItem['bookId'] == $bookId) {
            $_SESSION['cartItems'][$bookId]['quantityInput']++;
            $found = 1;
            break;
        }
    }
    if ($found === 0) {
        // your query code here
        while ($book = $result->fetch_assoc()) {
            $newCartItem = array_merge($book, ['quantityInput' => 1]);
            $_SESSION['cartItems'][$book['bookId']] = $newCartItem;
        }
    }
}

Note several things about the refactored solution:

• I eliminated the check to see if the cart was empty or not. That is really not relevant to your case. All you need to think about is checking if the book exists in the cart. Whether it doesn't because it's empty or simply because other books are in there is all the same for your logic - you just need to append a new item.
• $_SESSION['cartItems'] ?? [] is the equivalent of isset($_SESSION['cartItems']) ? $_SESSION['cartItems'] : [] and will tell the loop to use an empty array if there is no cart, which will in turn just skip the loop and go straight to adding the new element.
• I eliminated the passing by reference and modified the session variable directly.
• When adding a new book, I renamed the variable from $cartItem to $newCartItem. I see it as a good practice not to reuse variable names, it can lead to unexpected results due to accidental reassignments.
• $newCartItem = array_merge($book, ['quantityInput' => 1]); does the same thing as your code did with array('bookId'=>$book['bookId'], 'title'=>$book['title'], 'price'=>$book['price'], 'quantityInput'=>1) - it uses the same keys and values of the book, just adding quantity to it.