I agree that where is better than that in your sentence. However, I'm unconvinced that the version in your question isn't "grammatically okay."
If you read between the lines, there are actually three questions buried in your question here:
1) Is the island where plants are green preferable to the island that plants are green?
Answer: Yes, I think most native speakers would find the version with where more natural and less jarring to the ear.
2) Is the island that plants are green grammatically okay?
Answer: This is where things get tricky, because the line between "grammatically correct" and "awkward" can get very fuzzy, and even depend on dialect. Just because a sentence can be improved doesn't mean the sentence is wrong in a grammatical sense. Whether that could be used instead of where in such a phrase depends largely on the rest of the sentence.
3) (from your title) Do native English speakers say like this?
Answer: Yes! Most people don't proofread their sentences before they are uttered, so native speakers will sometimes be heard saying things in a less-than-ideal way. It can be hard to find written instances of an awkward phrase, because such phases often get fixed before they get published. But that doesn't mean you would absolutely never hear something like this in an interview somewhere.
Moreover, sometimes which word you might use depends on the phrasing. For example:
- Hawaii is a place where the cost of living is high.
- Hawaii is a place that has a high cost of living.
Although I would generally use where in the first sentence, I couldn't use where in the second sentence, because the phrase "a place where has" simply doesn't work. So, even though the two sentences are similar, one uses "that" while the other uses "where".