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"Maybe I'm missing something, but I thought the first and second chapters of Acts referred to different events. The 120 were only at the first event"

You are missing something. Chapter 2 : 1,4 "they were all" the same as Chapter 1:13 "And when they were come in, they went up into an upper room, where abode both Peter, and James, and John, and Andrew, Philip, and Thomas, Bartholomew, and Matthew, James [the son] of Alphaeus, and Simon Zelotes, and Judas [the brother] of James. These all continued with one accord in prayer and supplication, with the women, and Mary the mother of Jesus, and with his brethren. And in those days Peter stood up in the midst of the disciples, and said, (the number of names together were about an hundred and twenty,)".

Interesting, James and Jude, Jesus' brothers and Mary's sons were there but not as Apostles. They are called disciples as were the women, so I assume based on your reading of this, since they were filled with the Holy Spirit and commissioned at the same time the Apostles were, they could baptize and give the Holy Spirit by the laying on of hands.

There is a good reason to think that these are two events, not one. That is because the continuity is broken by "And when the days of the Pentecost were accomplished, they were all together in one place" (Acts 2:1) If the narrative still placed them in the Upper Room, there would ne no need to time and place reference.

But that is not really my argument. Of course, the Holy Ghost can appear at any time to anyone, as well as Christ, the Blessed Virgin or the saints. When it happens it is a miracle of God. The scripture, however, also records how things are ordinarily: the Holy Ghost was given to a select group, some of which became priests and bishops. They proceeded to baptize and teach others, who received the Holy Ghost through their mediation. This practice continues to this day.