What is the reason why に is used in this sentence instead of が or を?
外国語を使って仕事をしていらっしゃる方々にお集まりいただきました。
Does this いただく here stand for honorific of する or of もらう?
I think it's from もらう here. 〜に〜してもらう is used to express that you "receive the action from someone". They do it for you and に is the marker for the person doing it.
いただく makes it humble, so you express that it's something like a favor for you.
お集まり is the honorific version of 集まる. As you are using いただく afterwards, you don't need the する anymore. It seems almost like a nominalization to me.
The whole sentence sounds like "We have assembled people here who are working in a foreign language", literally "We humbly received (the favor of) assembling from people who are working in a foreign language".