I would look for
1. Freedom from all typos, misspellings, and careless use of punctuation, first of all.
2. Clarity of expression.
3. Grammatical accuracy (but probably not to a standard of absolute perfection).
4. Appropriate style and tone.
5. Evidence of intelligent thought.
6. Facility.
I also zero in on certain kinds of mistakes. I watch for unevenness, which suggests (but does not prove) a helping hand, and wrong word choice, which indicates a poor command of the language in general.
The ideas expressed also tell you something about the person, perhaps more than the writer realizes. I would read between the lines and infer something about personality and character.
If you would be in charge of the public face of a campaign online, they will need to know that you won’t embarrass them, that you will make them look good, and that you won’t introduce errors.
Papers on some academic subjects might show your skills to better advantage than others, but any will show gross deficiencies if you have them. Depending on how many applicants they have and where you are in the process, the writing sample could be used to eliminate candidates rather than to select them, so avoiding negatives could be more important than exhibiting positives.