Straight answer: a browser cannot tell you, and it is worth knowing why so you stop looking for a site that will.
The Gamepad API exposes a device identifier string, the button and axis counts and the input values. That is deliberately all of it, for privacy reasons: the less a web page can read about your hardware, the less it can fingerprint you. Firmware version is not in that set, and no browser exposes it. Any site claiming to read your controller firmware is either guessing from the identifier string or is not doing what it says. Our firmware version check tells you where to look and what current versions are, rather than pretending to read it from the page, and that is an honest limit rather than a missing feature.
For your Xbox controller there are two places. On Windows, install the Xbox Accessories app, connect the controller by USB, and it displays the firmware version and tells you whether an update is available. On a console, go to the accessories settings, select the controller, and the version is listed there. Both are reliable and both work without changing anything, which is what you asked for.
For the other common controllers, so you have it in one place. A DualSense does not display a version anywhere in the PS5 interface; the console simply prompts you when an update exists, and Sony's PC support tool reports it on Windows. A Switch Pro Controller or Joy-Con shows controller firmware state through the Controllers screen, which offers an update when one is pending rather than listing a number. Third party pads vary completely and most require the manufacturer's own utility.
One practical note on your underlying question. Being behind on firmware is not automatically a problem, but controller firmware updates do fix real connection and input bugs, so if you are several versions behind and have any dropout or input issue, updating is one of the highest-value things you can do and it takes a couple of minutes.
