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How can I make, for example, Stockfish 13 or 14 with a NNUE neural network? What is the precise procedure for making a NNUE neural network for some chess engine? I noticed that lichess.org uses Stockfish with NNUE and I want the same in with my Stockfish engine. How does one go about doing this?

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For Stockfish there are two options - you can either use an exe with an embedded network, or download the network separately and place it in the same folder as the exe.

In either case, you will have to enable the NNUE option. If you are using the command line UCI protocol, the command will be:
setoption name Use NNUE value true

If you are using a GUI like chessbase, there will be a checkbox somewhere that you can set:

Chessbase enable NNUE Note that I have downloaded an explicit NNUE file, and the relative path is set in the EvalFile option

The Stockfish GitHub page has a few more details as well:
https://github.com/official-stockfish/Stockfish#a-note-on-classical-evaluation-versus-nnue-evaluation

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    I'm not sure if this is the question that was asked - it sounds like Mr. J wants to know how to train a new NNUE for Stockfish.
    – Allure
    Aug 4, 2021 at 0:53

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