How do I automatically execute an external command in Vim/Neovim?

This can be achieved using an autocmd:

autocmd BufWritePost *.py !black %

It will run the your command after a file with the extension .py has been written to file. For more information see :h autocmd.

"Automatic exeuction" in Vim/Neovim is related to the concept of auto-commands. TJ DeVries (the author of Lua Autocmds in Neovim) has some great videos on his YouTube channel, specifically the Autocmd Intro, Autocmd Groups and Autocmd Patterns videos.

For a Neovim/Lua version of the auto-command that does what you are wanting to do (and ensuring that one properly assigns the auto-command to an auto-command group to prevent it from being loaded repeatedly every time a file is re-sourced):

local autocmd_group = vim.api.nvim_create_augroup(
    "Custom auto-commands",
    {clear = true})

vim.api.nvim_create_autocmd({"BufWritePost"},
    {
        pattern = {"*.py"},
        desc = "Auto-format Python files after saving",
        callback = function()
            local file_name = vim.api.nvim_buf_get_name(0) -- Get file name of file in current buffer
            vim.cmd(":!black --preview -q " .. file_name)
            -- vim.cmd(":!isort --profile black --float-to-top -q " .. file_name)
            -- vim.cmd(":!docformatter --in-place --black " .. file_name)
        end,
        group = autocmd_group,
    })

I'd recommend installing isort and docformatter as well. You can then uncomment the two lines above so that imports and docstrings are formatted and organised consistently.