May a child be trusted to immerse a vessel?
A child [who is below Bar/Bas Mitzvah] may not be trusted [alone] to immerse a vessel in the Mikveh. However, if the child immerses the vessel in the presence [and under the supervision] of an adult [male or female above Bar/Bas Mitzvah], then the immersion is valid. This applies even if the child is not educated in the purpose or meaning behind the immersion, and simply performs the act of entering the vessel into the Mikveh. [In such a case that a child immersed the vessel under supervision, the adult supervisor must inform the person for whom the vessel was immersed, that the child immersed the vessel properly. Whenever a child immerses under adult supervision, he is to recite a blessing over the immersion.]
Glass vessels, and all vessels which require Rabbinical immersion: Some Poskim rule that the above invalidation of a child immersing a vessel without supervision, only applies if the vessel requires Biblical immersion, such as if it is made of metal. However, if the vessel only requires Rabbinical immersion, such as if it is made of glass [or plastic] then a child who has reached an age of mental understanding may be trusted to immerse it even without adult supervision. Other Poskim, however, rule that a child may never be trusted without supervision even if the vessel is made of glass.
One may not send a child below the age of Bar/Bas Mitzvah to immerse one’s vessels. It is best to abstain from doing so even with vessels of non-metal materials, such as glass. It is, however, permitted for the child to immerse the vessel under adult supervision, and have the child say the blessing. Thus, one may take his child with him to the Mikveh Keilim and have him immerse the vessels in the Mikveh while one is watching. Likewise, one may send the child to immerse the vessels if he prearranges for an adult to supervise him and report that the immersion was done correctly.
May a child who is Bar/Bas Mitzvah be trusted even if one does not know if he has grown the signs of puberty?
Some Poskim rule a child who is above Bar/Bas Mitzvah may be trusted even if one does not know that they have grown signs of adulthood [i.e. two pubic hairs]. Other Poskim, however, rule that the child may only be trusted to immerse Rabbinical vessels, such as glass, but not by metal.
May one trust a child to give the vessel to an adult and have the adult immerse it?
Yes. A trustworthy child may be entrusted to do so if he returns and testifies that he did as asked, and an adult immersed the vessel.
 Yoreh Deah 120:14
 Michaber ibid; Terumos Hadeshen 257; Kitzur SHU”A 37:12
The reason: As the Mitzvah of immersion is a Biblical obligation, and by all Biblical obligations, a child may never be trusted. [Taz 120:16; Terumos Hadeshen ibid; Beir Hagoleh ibid; Admur 432:10; 437:6 regarding trusting a child to perform a Biblical Bedikas Chametz and Admur Y.D. 1:42 regarding trusting a child to slaughter] The following Poskim rule that Tevilas Keilim is a Biblical obligation: Stam opinion in Admur 159:21; 2nd opinion in Admur 323:8 and final ruling there; Implication of Michaber 320:9 and 14 and 17 who rules a Katan is not believed; Taz 120:16; Peri Chadash 120:1 [see there in length]; Rashba in Teshuvah; Raavad; Ramban; Rashi Avoda Zara 75b; Tosafus Avoda Zara ibid; Rabbeinu Tam Yuma 78; Semak 199; Issur Viheter 58:91; Aruch Hashulchan 120:4 that so agree majority of Poskim
Other opinions: Some Poskim rule the Mitzvah of immersing vessels is a Rabbinical obligation. [1st opinion in Admur 323:8; Rambam Machalos Assuros 17:5, as understood by Rashba in Toras Habayis p. 155 and Peri Chadash ibid [however see Rashba in Teshuvah ibid and Aruch Hashulchan 120:1-3 for alternative understanding in Rambam]; Ran; Ritva; Or Zarua 1:359; Radbaz 34 that so rule majority of Poskim; Yeshuos Yaakov O.C. 509:4] Accordingly, a child who has reached the age of Chinuch, may be trusted to immerse the vessel. [Implication of Taz ibid; Terumos Hadeshen ibid; Admur 432:10; 437:6 regarding trusting a child to perform a Rabbinical Bedikas Chametz]]
 Rama ibid; Terumos Hadeshen 257; Misgeres Hashulchan 37:9
 Implication of Michaber and Rama 120:15 who validate the immersion of a gentile; Shach 120:28 and Taz ibid that so is implied to be the final ruling of Rama who does not make this condition, and so is their final ruling; Levush, brought in Shach ibid; 2nd answer in Terumos Hadeshen ibid and so is his conclusion
The reason: As Tevila from impurity to purity does not require intent at all. [Poskim ibid; Teshuvas HaRashba and Ramban 151, brought in Bach 120]
Other opinions: Some Poskim rule, one is to educate the child in the purpose of the immersion, that it is done to purify the vessel, otherwise, the immersion is invalid even under supervision. [Bach 120, brought in Shach 120:28; 1st answer in Terumos Hadeshen ibid; Biur Hagra 120 based on Rama 198]
 Darkei Teshuvah 120:105
 1st approach in gloss of Rav Akiva Eiger 120:14 and that so holds Shagas Aryeh; Panim Meiros 2:96; Aruch Hashulchan 121:13; Chochmas Adam permits to be lenient for the sake of Shabbos, brought in Misgeres Hashulchan 37:9; Implication of Shach 127:31 in name of Rivash that a child is trusted even if there is Chezkas Issur if it is Beyado; Implication of Erech Hashulchan 115 in name of Radbaz Implication of Admur 432:10 in parentheses and M”A 437:8 regarding trusting a child to perform a Rabbinical Bedikas Chametz even though it is Ischazeik Issur and concludes “And anything which a child can do, he is believed to say he did it as explained in Yoreh Deah 120 see there” , however see Admur Y.D. 1:42 that a child is only believed by matters that have no Biblical root.
Opinion of Admur: Admur Admur 432:10 in parentheses rules that we always believe a child by a Rabbinical matter that is within his hands to perform, even if there is a Chezkas Issur. This would imply that Admur rules like the Poskim ibid that a child can be trusted to immerse a glass vessel. However, Admur Y.D. 1:42 states that this only applies if the obligation does not have a root in the Biblical law. Hence, a child would not be trusted to immerse a glass vessel, as the concept of Tevilas Keilim is Biblical. Vetzaruch Iyun!
 Peri Chadash 120:3 and 26; Gloss of Rav Akiva Eiger 120:14; Shagas Aryeh; Panim Meiros 2:96; Chochmas Adam 73:19, brought in Pischeiy Teshuvah 120:3; Aruch Hashulchan 120:24; Rashi, brought in Aruch Hashulchan 120:1 “Specifically metal”; See Taz 120:1 “It is only common to use for a meal vessel made of metal.”
 See Admur ibid and Michaber 437:4 that we only trust a Katan “Sheyeish Bio Daas”
 Peri Megadim 451 M.Z. 7, brought in Pischeiy Teshuvah 120:14; 2nd approach in gloss of Rav Akiva Eiger 120:14; Ben Ish Chaiy Matos 2:9; See Rama Y.D. 127:3 that we only believe a child by Bedikas Chametz because there is no Chezkas Issur.
 The reason: As a vessel contains a Chezkas Issur, and a child is never believed to testify by a chezkas Issur, even if it is Rabbinical. [P”M ibid; See Rama ibid] Alternatively, as even by Rabbinical matters, a child is only believed by actions that he has responsibility to do for himself, such as to immerse a vessel that he owns, as opposed to matters which he performs as a Shliach, on behalf of others. [Rav Akiva Eiger ibid based on Tosafus Eiruvin 31b] These two reasonings are negated by Admur and M”A ibid regarding Bedikas Chametz, as they hold that whenever one has ability to fix it himself, a child is believed even by Chezkas Issur
 Pischeiy Tehsuvah 120:14; This is known as Chazaka Derava. A child is not Halachiclly considered obligated in Mitzvos until they have grown two pubic hairs. Rava held that by Rabbinical matters one may assume that they have grown it, and they may thus be trusted to be Bar/Bas Mitzvah, however, by Biblical matters they are not trusted. See Rava Nida 46a; Admur 39:1; 53:9; 55:6; 128:49; 199:9; 271:7
 Pischeiy Teshuvah 120:14; Rav Yitzchak Elchonon in Zecher Simcha; Teshuvos Vehanhagos 3:405; See Hakashrus 4 footnote 108
 The reason: As since using a vessel prior to immersion does noty prohibit the food, therefore, it is given a lesser severity than a regular Biblical obligation, and one may adapt the Chazaka of Rava. [Poskim ibid]
 P”M 451 M”Z 6, brought in Pischeiy Teshuvah ibid
 See Admur Y.D. 1:42 regarding Shechita “Nonetheless, the custom is to send a child to give the animal to a Shochet to slaughter, and we do not suspect that he will do so himself.”; Chanoch Lanaar 30 footnote 2, brought in Hakashrus 4 footnote 39
 The reason: As a) There is a Chazaka that a child will not do things to stumble others. b) He fears telling a lie. [See Admur ibid]