SOLUTION: To solve the problem simple upload the public/build folder in the private/laravel/public/ of your web hosting. as the error is looking for the manifest.json file in the path as described in the error.

Exception
PHP 8.0.27
9.32.0
Vite manifest not found at: /home/user45/web/hosting_account/private/laravel/public/build/manifest.json

Missing Vite Manifest File
Did you forget to run `npm install && npm run dev`?

app.blade.php

<!DOCTYPE html>
<html lang="{{ str_replace('_', '-', app()->getLocale()) }}">
    <head>
        <meta charset="utf-8">
        <meta name="viewport" content="width=device-width, initial-scale=1">

        <title inertia>{{ config('app.name', 'Laravel') }}</title>

        <!-- Fonts -->
        <link rel="stylesheet" href="https://fonts.bunny.net/css2?family=Nunito:wght@400;600;700&display=swap">

        <!-- Scripts -->
        @routes
        @vite('resources/js/app.js')
        @inertiaHead
    </head>
    <body class="font-sans antialiased">
        @inertia
    </body>
</html>