According to all halachic authorities, an adult may allow a Jewish minor who has reached the age of Torah education (seven or eight years of age) to immerse a utensil under the adult’s supervision. In such a case, the minor may even recite the berachah before the immersion.
If, however, a minor claims to have immersed a utensil without adult supervision, then, according to the Ben Ish Chai (a”h), the child is not trusted and the utensil still requires immersion. Ha’Rav Ovadiyah Yosef shlit”a, however, distinguishes in this regard between metal and glass utensils. With regard to metal utensils, which require immersion on the level of Torah obligation, one should not trust a minor who claims he immersed it. Glass utensils, however, require immersion only by force of Rabbinic enactment and one may therefore trust the child’s claim and the utensil need not be immersed before use.
As for non-Jews, one should preferably not allow them to immerse his utensil on his behalf, because he would thereby forfeit the recitation of the berachah. One cannot recite the berachah over the immersion if a gentile immerses the utensil. Therefore, in order not to forfeit the opportunity to recite the berachah, one should preferably immerse the utensil oneself. However, if a gentile did immerse one’s new utensil in a Mikveh in front of the Jew, and even if the utensil fell into the Mikveh by itself, the immersion is effective and the utensil may then be used without an additional immersion.