How do I make "X is the thing I wanted to win" unambiguous?
Solution 1:
Out of context the sentence StackExchange is the website I wanted to win is inherently ambiguous because of the nature of the verb to win, which is ambitransitive. In other words, win can be both transitive and intransitive:
She won the award. (transitive)
She won. (intransitive)
So if Polly is a cat and I read the decontextualised sentence Polly is the cat she wanted to win, I cannot know if she wanted to win Polly (transitive), eg. in a competition to win a cat. Or if Polly is a cat that she hoped would win in a Best Cat competition.
The ambiguity does not arise in similar constructions where the verb has only a transitive meaning:
Polly is the cat I want to have.
ELL is the site he likes to consult.
Your daughter is the child you need to hug.
Polly, StackExchange and your daughter are objects of the transitive verbs to have, consult and hug; they cannot be the subjects of those verbs.
Nor does ambiguity arise with intransitive-only verbs in these (somewhat contrived) sentences:
Mary is the girl I want to sit.
StackExchange is the site I least want to die.
Mary and StackExchange are the subjects of sit and die. They cannot be the objects of these intransitive verbs.
I have not been able to think of a particularly plausible example of a sentence containing an ambitransitive verb that is likely to be ambiguous in the way that Polly is the cat she wanted to win is. Possibly:
The sax is the instrument she wanted to play
which could be construed as She wanted to play the sax (transitive) or She wanted to hear the sax play (intransitive) - as the next instrument in an improvised jazz piece.
As several of the comments above indicate, the context is almost always sufficient to disambiguate meaning. But in this case, adding the verb to see, as suggested, makes the meaning absolutely clear:
StackExchange is the website I wanted to see win
Solution 2:
Although there is technically ambiguity in your original phrase, the context makes it very obvious which meaning you meant. However, if you are really desperate for a phrase with no ambiguity I would go for:
I wanted Stack Exchange to win
Solution 3:
There are a number of ways you could structure this sentence, all of which would convey your point precisely as you mean it. Firstly, you could say "I hope that StackExchange wins [the Best Website Award]". This implies that, of the various potential 'nominees' for the Best Website Award, you want StackExchange to win the award. Alternatively, you could say "In my opinion, StackExchange should win [the Best Website Award]." This implies the exact same thing as the previous example, but makes the statement seem more opinion based. I have included the Best Website Award in brackets so that if you do need to be more explicit in your statement, you can use the term.
Solution 4:
The second way you phrased it in the question was clear:
"I wanted StackExchange to win [the best website award.]"
The bit in square brackets could be omitted if clear from context, to satisfy your requirement "without explicitly mentioning that it's the award that I want them to win."