Can you apply for a faculty position via a recruiting agency?
I have never, ever, encountered a faculty search - or indeed a postdoc search - handled by an agency.

Nor have I ever been contacted by an agency looking to handle recruitments as a potential candidate, despite getting ads on sites like LinkedIn that suggest I've been accurately categorized as faculty.

In the vast universe of possibilities, I'm sure there are exceptions, but my best guess is that this is boiler-plate language that goes in all job ads from that institution.