Do I have to say the intention out loud for wudu?

No. The intention (niyyah) for wudu is in the heart. If you stand at the sink with the aim of purifying yourself for prayer and then wash, your intention is complete. You do not have to say anything out loud, and wudu is valid without special words.
Intention belongs in the heart
Niyyah simply means to intend, to mean what you are doing and why. The Prophet ﷺ said that deeds are judged by their intentions and that everyone gets what they intended (Sahih al-Bukhari 1). Wanting to make wudu in order to pray, and then doing it, is the intention.
The Quran also describes sincerity as an inner matter, directed toward Allah alone:
What scholars say about speaking it
There is no report that the Prophet ﷺ or his Companions pronounced a set phrase before wudu. For this reason, most scholars say it is not needed and not part of the sunnah.
- Maliki, Shafi'i and Hanbali schools: the intention is required for wudu, but it is a matter of the heart, and speaking it aloud is not a condition.
- Hanafi school: intention is a sunnah for wudu, not a condition for its validity, though it is needed for tayammum.
- Some Shafi'i scholars: quietly saying it can be recommended to help focus the mind, but even they agree the heart is what counts.
If saying the intention helps you concentrate, it does no harm, but it should not be treated as a required act or a special Arabic formula.
What counts as an intention?
You don't need to produce a feeling or a thought each time. If you decide to make wudu, go to the sink and wash with the purpose of purifying, you have made the intention. Habit doesn't remove it. See How do I make wudu? and What are the obligatory parts of wudu?
If worrying about whether you "meant it" keeps coming back, treat it as a whisper and don't act on it. See How do I deal with waswasa in wudu and ghusl?
What you can say instead
The Prophet ﷺ's practice at the start of wudu is to say Bismillah, and you may make a short dua at the end. See Do I have to say Bismillah before wudu?
- The intention for wudu is in the heart; no words are needed.
- Deciding to make wudu for prayer and then washing is the intention.
- Most scholars say speaking it is unnecessary; some Shafi'i scholars allow it quietly.
- Don't let doubts about your intention turn into repeated wudu.
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