dbwcbb- you can definitely ask the shadchan to meet the girl your son is dating.
I was asked to meet mothers of a few of the boys I dated, but it was always after 2-3 dates. It wasn’t a big deal, we usually met in a far-out Starbucks and shmoozed for 10 minutes.
After my fourth date with my husband, he took me to meet his parents at his house. A little awkward, but again not a big deal.
Good luck!