If you want only the user_id then it is a property on each Order_model instance in the Collection. If you want the User instance, then you need a second relationship on the Order_model model. It is difficult to understand really what you want/need however.
Your code is needlessly verbose, and not expressive; it is difficult to understand exactly what you are trying to retrieve because your variable names are inconsistent, and you are making multiple queries unnecessarily for the same User instance as auth()->user() already is; and you have an uncompleted Eloquent Query for $subcategory. For example, a simple refactor removes 4 such queries:
public function vendor_order()
{
$cart = auth()->user()->cart;
$messagesCount = auth()->user()->latestmessaging->count();
// $slug = User::where('slug', auth()->user()->slug)->firstOrFail(); // this is auth()->user()
$subcategory = SubCategory::with(['productCategory']); // this is not completed
$order = auth()->user()->orders;
return view('user.vendor_order', compact('cart', 'messagesCount', 'subcategory', 'order'));
}