In this structure:
"They are getting more and more applicants wanting to join them".
What is that? Is it the participle? Is it the "get sb doing" clause?
As always in English, there's probably more than one way to parse that, but this is what I think makes most sense...
I would say that "wanting to join them" is an attributive verb phrase. That is a non-finite verb phrase used to describe a noun - in this case the applicants. One of the attributes - that is to say, characteristics - of the applicants is that they "want to join them". That might seem tautological, of course the applicants want to join them or they wouldn't be applicants. However, it might be a necessary clarification, as people apply for things other than 'joining' something (they apply for loans, or grants, or driving licenses). It might also be for emphasis, or to make clear that what's interesting is that more people want to join them.