Can you purchase the techelet dye from a gentile? Can you purchase tzitzit that have been dyed with techelet by them, is that valid?
התכלת אינה נלקחת, אלא מן המומחה--חוששין, שמא לא נצבעה לשמה.
Tekhelet may not be purchased but from an expert, for we are concerned lest it not have been dyed for the sake of [the commandment]. (my translation)
Whether or not a gentile who makes an item for the sake of a commandment with a Jew standing there telling him to do so counts is a dispute between Rambam and Rosh (OC 11:2 etc.) though many later authorities advise being strict.
המפקיד תכלת אצל הגוי--הרי זו פסולה, שמא החליפה. ואם הייתה בכלי, והיה חתום בשני חותמות, חותם בתוך חותם--כשרה; בחותם אחד, פסולה.
One who deposits Tekhelet with a gentile -- it is invalid, lest he exchanged it [for a cheaper fake]. And if it was in a vessel sealed with two seal, one seal within the other seal, it is valid. With one seal, it is invalid. (my translation)
If buying Tekhelet from a gentile you would need careful supervision and packaging to ensure nothing has been tampered with.