Collecting and expanding on my comments on the question:
Do university departments take on additional faculty members for existing lab groups?
TL;DR: no, but faculty do sometimes hire Ph.D.s into their research groups.
my focus is primarily on research groups in the US.
And so is this answer.
By 'lab group', I mean a small-ish group of researchers (~5-15) who
are collaborating together in a fairly specific sub-field. It could be
theoretical or numerical, I don't mean to imply that they are
necessarily doing experimental research.
You seem to be coining the phrase "lab group". At least, it is not one with which I am familiar.
What you describe is similar to what I know as a "research group", and I recognize that you used that terminology in an earlier revision of the question. From comments, I think that's indeed what you mean, but in that case, the question belies an unfamiliarity with the conventional organization of academic research that seems surprising in someone approaching completion of their Ph.D..
I suspect that's among the reasons that your question has been difficult for many of us to understand. No, faculty are not ever hired into existing research groups, because at academic institutions where faculty are expected to perform research, they are expected to do so by forming and leading their own research groups. The ability to do so is one of the criteria on which such institutions evaluate faculty candidates. Running an independent research program is one of the defining characteristics of being faculty at a research university.
Thus, when you ask about faculty being hired into existing "lab groups", we're primed to think that you mean something other than "research group". In fact, in the laboratory sciences, it's relatively common to use the term "lab" interchangeably with "research group", so a "lab group" could be interpreted as some kind of higher-level group of multiple faculty members' research groups. And collaboration among research groups is very much a thing -- often, even, across multiple institutions.
Alternatively, the term could be interpreted in the sense of access to physical laboratory facilities. It is indeed common for research institutions to share some equipment, especially expensive equipment, among multiple research groups.
But neither of those interpretations seems to be what you mean, so going back to "research group": no, faculty are not hired into existing research groups because that's inconsistent with being faculty. However, it's fairly common, at least in the sciences, for faculty to hire post-doctoral researchers into their research groups. Postdocs work under the direction of the supervising faculty member, but they generally have more independence than graduate students. Perhaps that's the kind of environment you're looking for. And in some fields, it's hard to get a faculty job without first doing a postdoc stint or two, which may be another reason to be looking in that direction.
Any way around, your Ph.D. supervisor is the best person to advise you here. No one else is as well attuned to both the norms in your particular field and your own personal strengths and weaknesses. Many faculty consider providing such career guidance to their students to be among their mentoring responsibilities, but even if yours doesn't consider it a duty, I certainly hope they would be willing to have such a conversation with you.