There's not any way to do that. It's okay not to say thank you (because there's not any way to do that). Just say it in your heart.
Keep in mind they deal with a big volume of requests, most of which are on the quick side. Acknowledging thank you's would either be 1) superfluous or 2) time consuming enough to actually impact their ability to work through the moderation queue. Keep in mind the point of even saying "thank you" when conducting business over the phone is to clearly say "I have no further business" all the moreso than to express gratitude, and that etiquette doesn't add value in this context.
If you really want to thank the moderator anyway, SE doesn't have private message channels. You could reach out to the moderator personally, if they have their email or site posted on their profile (most do I think). I'm taking neutral stance on recommending or suggesting this - if a moderator's contact info is there it's presumably so people can use their own judgment in contacting them, and thanking someone for something trivial could come across as any of ignorant or annoying, poignant and human, or perhaps be the lead you need to go into business together some day. That's what networking is, and it's not my business to say whether you should or whether we encourage this; it's just a physical option you have.