Do I need to code value labels in Qualtrics?

I mean, you don't have to change the value labels in Qualtrics. I often just work with the choice text instead of the value labels, especially with categorical questions like that one.
In your example, the value labels automatically assigned by Qualtrics would not match your codebook. Qualtrics will generally assign value labels to your questions incrementally; any new response item added to your question will get a value label one higher than the previous item. So in your example, "Some other race" would probably get a value label of 6, not 15. Similarly, "Prefer not to answer" would get a value label of 7, not 99. I think this is generally fine unless you actually care about the value labels and are trying to use the value labels in some way, like for scoring. If you don't have to use the value labels, then it doesn't really matter. If you do want to use the value labels for whatever reason, then you will need to edit them somewhere - be that Qualtrics, Stata, R, or whatever else you're using.
If you do want to use the value labels for whatever reason and need to edit them, then yes, I would choose to change the value labels directly in Qualtrics. I think that will just make your life a little easier later. But that's definitely more of a personal preference than a necessary step. Since it can be done in the code, it's not critical to get it perfect in Qualtrics.