Can I perform sujud shukr without wudu/ablution? At any state, and at any time?
The major Issue with sujud a-shukr سجود الشكر (or prostration of gratitude) is that scholars have different opinion on how to classify it (note the same sitatuion is with sujud at-tilawah, the sujud if you read a verse which orders a sujud in the Quran): some say it's (like) a nafl prayer some say it's a pure sujud only and not a prayer at all.
Sujud a-shukr is well evidenced by ahadith from Jami' at-Tirmdihi, sunan abi Dawod or sunan ibn Majah and by narrations of what the sahaba (May Allah be pleased with them) used to do, for example the prostration of abu Bakr when he got the message that Musaylima al Kadhab has been killed ...
Scholars who regard it a kind of nafl prayer (the shafi'i school, most of the hanbali scholars, some hanafi and maliki scholars) say it needs anything a prayer would need, niyyah (intention), tahara (purity), perform it in the direction of the qiblah, satr al-'awrah (hide the 'awrah) etc.
Those who say it's only a sajdah (without relation to prayer) (some maliki scholars, ibn Jarir at-Tabari, ibn Hazm, ibn Taymiyyah, ibn al-Qayyim, a-Shawkani and a-Sna'ani and most modern representatives of the salafi school) say it neither needs tahara nor anything which might be necessary for a prayer. Most fatwas I could find are saying that this opinion is stronger. As the person who reads or recites the Quran from his/her memory doesn't need to be purified (by ghusl or wudu) so if he or she was not allowed to perform a sujud at-tilawa he/she would miss lot of thawab or rewards, so logically this can't be the case. This can also be concluded from the hadith from above and the fact that our Messenger (Peace be upon him) didn't say anything about that.
So according the favored opinion you neither need wudu' nor any kind of purity, you don't even need to face the qiblah, you don't need to wear hijab or hide your 'awrah, it's not sunnah to say "Allahu akbar" at the beginning nor at the end and no tashahud or salam is needed according the shafi'i and hanbali madhab.
And Allah knows best.
The correct ruling according to Qur'an and Sunnah is that sajda shukr/sajda tilawat do not require wudu. Abdullah ibn Umar used to do sajda tilawat without wudu. Not every scholar heard about this narration so they were in doubt, that's why some say that it's required. In reality, no, wudu is not required.