The requirement is only by standards. It is perfectly possible to do whatever you want on a page and not follow standards. Things may not display or work correctly if you do that, but likely they will. The goal is to follow them, and the idea is that if you follow them, your page will always work; you don’t have to worry about anything.
Yes, the form is required to have an action attribute in HTML4. If it’s not set, the browser will likely use the same method as providing an empty string to it. You really should set action="" which is perfectly valid HTML4, follows standards, and achieves the same exact result.
In HTML5, you can actually specify an action on the submit button itself. If there isn’t one, it uses the form’s action and if that is not set, it defaults to the empty string (note you cannot explicitly set the action to an empty string in HTML5).