No, poor writing as a student is not necessarily an impediment to becoming a good researcher. You will just need to work hard to improve and find a good advisor or fellow researcher to critique your writing.
I think that good writing is linked with good thinking. I have spent a lot of time reading work by non-native English speakers. Sometimes, even if the grammar and word usage is wrong, I can easily tell what the writer is trying to say because their writing is logical and well-organized. I have a colleague whose mother tongue doesn't have articles. He will probably miss some necessary "a"s and put "the"s where they are not needed for the rest of his life, but this is quite easy for any native speaker to clean up since his arguments are always very clear. Writing style is less important than logic and organization and, for students early in their career, this is often the biggest problem. This sometimes has nothing to do with writing style and everything to do with the fact that they don't really have a broad understanding of the research field they are writing about: they include many small irrelevant details and do not make the necessary big picture arguments. Also, they often put no thought into organization and just spill out the ideas in a random order with no links between them.
Of the several graduate students and postdocs I've mentored (~15), almost all started out as what I would call poor writers. They all improved through writing their own papers and going through many drafts with me. I suggest that you find an advisor who you consider to be a reasonably good writer and is willing to go through your papers with you, making specific comments on how you can improve. If your advisor won't help you, see if you can find another researcher willing to help. Talk to other students in the group and find out if they are good at mentoring their students through writing.
Writing is absolutely essential for an academic research career, and after becoming relatively senior you will spend most of your time writing.
If you are a non-native English speaker, you may think you are a poor writer because people correct mistakes in your grammar or word usage. I wouldn't worry too much about this because these things can be superficial. Focus on a logical argument and good organization, and you can probably find a coworker (or pay someone) to fix your English.