This very much sounds like a spam email from a person who wants to work on your client's SEO for a lot of money. I receive those continually, at times even from websites that are gone.
As for whether it did happen, if you have Google Search (you can create an account now, though, and get some info from the past) then you can see how many searches you had and see whether it indeed dramatically changed in Dec/Jan. (although I'm afraid the data doesn't go that far back anymore.)
Google Search is what shows you the organic results from Google. There is Bing Search too for Bing & Yahoo!.
A system such as Google Analytics would also show you other organic sources such as Social Media.
Now, watch out because:
(1) there are websites that are seasonal, you clients may be getting more traffic in the summer, so a slowdown around Dec/Jan would be normal
(2) there are daily/weekly/monthly patterns, I have a site that gets a lot of traffic from Monday to Friday and nearly nothing over the weekend
(3) there could have been a page that was getting a lot of traffic even though it was not producing any income for your company, just because a really influential person posted that on their social media and many users followed the link... this is most often a fad and it does not related to having good targeted traffic.
However, all of that would not allow anyone else to know about your traffic. There are only two ways for them to know: they have access to your analytics, they are part of a system such as AdSense and collect that data for themselves. Anyone else can't really know (I still don't really understand how amazon does it with their Alexa statistics, although they probably have access to some routers and gather the data from those directly...)