Short version, There are nonselective organic herbicides. do they work? not well but it makes people feel good.
Grumpy farmer version. Since your in texas things may be different down there. The term 'organic' doesn't mean herbicide or chemical free. It means that the chemicals that you are allowed to use must be naturally sourced product. As an example, if you squeeze a snake and get an oily product, that snake oil could be used as an organic herbicide. How ever if you were to add a man made substance to that product you couldn't sell it as organic. There are several 'organic' non selective herbicides that are available. In my professional opinion, I'm a farmer, the organic products aren't much better than snake oil, and because they are organic they don't fall into the same regulation guidelines that conventional products do the exact impact on the environment isn't know. Some studies show that organic products are more harmful to the environment than conventional chemicals. This is mostly due to the fact that the organic's aren't as potent and require many more applications to do the same job as one application of a conventional product. I'll stop my rant here. I could go on for days about how the entire organic movement was started and continues to be supported by a group of self righteous feel good hippies who think they are actually doing something good for them selves and the environment. When actually the whole organic movement is really a farce, and is actually worse for the environment and doesn't produce near the quality or quantity that modern farmers do.
Personally I'd tell them to apply the tordon or remedy, and wait the 4 years it takes to get the organic certification. Trees and crap like that aren't friendly to any ag equipment. Or they could use a product like AXXE or Interceptor, which is an organic certified product, in canada that is, and when no one is looking they could add some tordon, remedy, round up, grazeon or restore.