I second it. "Ty chceš vždy vyhrát" means exactly "always winning", not "always wanting". I believe Czech is flexible enough to say "Ty vždy chceš vyhrát." if it wished to convey "always wanting".
In the case of the prompt, however, the correct translation should stress the "always winning" aspect, so either "You want to win always", as suggested by Nina, or "You want to always win".
My question here is with the use of “vždy”. Could “stále” be used here as well? In these sentences regarding winning and losing I see that they are both used. My understanding is that “stále” is used as always in the continuous sense and that “vždy” is used as always in the repeated sense. Are they sometimes interchangeable? I’m never sure which one to use.
The issue with "stále" (and its synonym "pořád") is precisely its continuous sense which interacts with the aspect of the verb. If you said "Stále chceš vyhrát." (imo the best word order for that), it would really mean "You still want to win.", as if no winning took place yet. If you want to preserve the sense of always wanting to win, then the aspect needs switching to "Stále chceš vyhrávat."
With "vždy" or "pokaždé" rather than "stále", the interference with the perfective aspect is not present and we could use both aspects to mean the same thing (without implying the lack of winning up to now).