Chrome Store does not allow extensions to download
Chrome Store does not allow extensions to be downloaded
The first sentence above treats download as intransitive.
I can't get this file to download intransitive
The second treats download as transitive.
I want to download transitive this file direct object.
The Chrome Web Store does not allow extensions to download intransitive from YouTube.
Google won't allow the file to be downloaded. passive/transitive
Your sentence with prevent is correct when the verb is intransitive:
The Chrome Web Store prevents extensions from downloading.
Your sentence with allow is incorrect when the verb is transitive (when you wish to express the meaning "get downloaded") :
The Chrome Web Store does not allow extensions that download.
If transitive, you must use a passive construction if you're omitting the person who does the downloading:
The Chrome Web Store does not allow extensions that are downloaded
from YouTube
The Chrome Web Store does not allow extensions that have been downloaded
from YouTube
But if the verb is regarded as intransitive, your second sentence goes from incorrect to unclear and ambiguous: are you referring to extensions that are themselves downloaded or to extensions that download other files?
The Chrome Web Store does not allow extensions that download from YouTube.