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There are several processes, many of them documented on that very page, that allow an organism to pass on certain characteristics that happened during its lifetime to at least the next generation. One other example of such a process is horizontal gene transfer, typically through viruses, but also sometimes in plants through grafting.

For an extremely simple example, if a female organism is exposed to significant doses of radiation, the DNA inside the egg cells it was born with will have a high chance of suffering mutations. If one of those eggs remains viable and later gets fertilized, the DNA of the offspring produced from it will be significantly different from the DNA that of offspring born before the radiation exposure.

There are other more subtle and more common processes, though. The most surprising one is that DNA methylation, a process in which environmental stimuli encountered in a living sexually mature individual affect gene expression, has been proven to be transmissible to offspring across several generations in certain species. This includes stimuli such as prolonged exposure to heat or cold, long-term starvation, and others. This is indeed a form of Lamarckism that has actually been proven to happen, though of course it involves much subtler changes than some of what Lamarck envisioned.