This is an important question. How are you approaching populating this drop down?
If you were refreshing the page, then you would not have the problem you are describing. But if you are not, so I assume what you mean by "existing form" is that you are plugging an HTML form into a PHP page and wanting to populate it after the radio button is selected.
If I am interpreting this correctly, you would need to use JavaScript or, more likely Ajax. PHP comes before any action you can take on your form because it is server-side. It runs and writes the page according to the information it has at the moment the page is opened. In order to change that information, you need to go back to the server and rewrite (refresh) the page. (or use Ajax)
If I am misinterpreting, and your radio button is on another page, then write it as anakadote suggested, and you should be fine.