It seems to be common knowledge that the Ramchal did not wear a beard (see here for instance). What sources are there, if any, that say the Ramchal, in fact, did not wear a beard?
The Ramchal's beardless face is referenced in several letters among the Italian rabbinic communities of the 18th century, all of which are part of the general controversy that escalated after the Ramchal's assertion of learning through a maggid and his teachings of kabbalah, both activities that led some to suspect him of being a secret follower of Shabbetai Tzvi. The history of the anti-Sabbatean campaign that R. Moshe Hagiz and others waged against him is carefully related in Elisheva Carlebach's work The Pursuit of Heresy.
R. Yeshayahu Bassan, one of the Ramchal's teachers, was one of the participants in the campaign, who became increasingly frustrated with his student. In 1730, after the Ramchal asked that he write to Hagiz in support, Bassan replied with his own worries, such as the fact that he was still single and the "fact that you do not grow your beard" (cf. Letter ר׳ משה חיים לוצאטו ובני־דורו #39).
In another letter, R. Yosef Ergas wrote to R. Bassan about the issue of whether the Ramchal was a legitimate kabbalist. He wrote that he heard from others that while the Ramchal was learned in kabbalistic teachings, he did not consider him a Hasid [in the pre-Hassidic-Movement sense, meaning piousness], since:
ושאלתי אותם אם הוא נשוי ואם הוא נזהר בטבילה של ע״ש ושלא לספר זקנו כלל אפילו במספרים - ועל כולם השיבוני שאינו מקיימם כלל
I asked whether he was married and whether he was careful to immerse in the mikveh on erev Shabbat, and whether he was careful not to cut his beard even with scissors - and for each I was answered no. (Letter #21)
Carlebach found an amusing set of testimonies from the rabbanut in Venice in 1734, regarding students of the Ramchal who had claimed to have found "a mirror with a black frame, a knife with a black handle, and a candle of black wax" in his room, strange objects that some in the community feared was evidence of magical practices. After inquiring further, the rabbanut heard the Ramchal's explanation: they were the instruments of his shaving equipment (Carlebach 233).
The first source saying that Ramchal “did not let his beard grow” was his early teacher and later critic Rabbi Yosef Ergas. See here for a translation of the quote: