What is the word for someone who initiates group social interaction?

Icebreaker (or ice breaker)

An icebreaker is commonly considered to be a social device or method the relieve shyness and social tension at a gathering of people.

But I don't see any reason you can't use the term to refer to a person who has the role to do the same thing. That person is, by definition, using icebreaking devices to accomplish his objective.

This is backed up by the definitions in Urban Dictionary:

Icebreaker (definition #1) actually uses the term in a similar manner:

The person that shows you around when you're new to a community;
called the icebreaker because they usually help avoid awkwardness by
starting small-talk. Person 1: Who's that new girl? Person 2: I have
no idea. All I know is that Mary is playing icebreaker for her.

and similarly for Ice Breaker (definition #3)

Someone who breaks the awkward silence. Some one who mentions
something that has a lot of tension around it.

I don't think anyone would have any problem understanding what you mean if you said you had hired an icebreaker for your meeting or party (unless you live on the Beaufort Sea, maybe). They would probably be intrigued with the whole idea when they do understand.