Hmm, I thought it might be like mine, a Cairn Terrier, since they and the Yorkshire Terrier are similar.
Mine has a long-ish, natural coat and I sometimes bathe him at home, though you're not really supposed to too often, and I let the groomer deal with getting out the undercoat and dead hairs (modified stripping). I use a good quality conditioning shampoo and let him dry naturally, then brush him with a nylon people brush.
But I took a peek at grooming for the long haired Yorkie and it may be more complicated than that.
https://thehappypuppysite.com/yorkie-grooming/
You have to be very careful getting knots out as it really hurts them. (I just cut them out if they occur.)
Why do you want to bathe him, does he smell or did he get into something? Lol. Is it possible to text the owner and ask if it's ok to bathe him? Or maybe you could take him to his regular groomer. Otherwise, I might wait!