I'm not too worried about this. The number of publicly available datasets of reddit posts , many of them hosted on kaggle or bigquery (both owned by google) is very large, which suggests to me that reddit don't mind about this in the same way that say, Twitter does. It was also a deliberate decision to not include the reddit usernames in this data, and I personally don't think this would be a great resource to try and break someones privacy, compared to what else is out there.
That said, if either Kaggle or reddit do have a problem with this I won't hesitate to remove it.
If someone deletes their post on Reddit it will still be stored and available on your site.