Copenhagen to Møns Cliffs

I'd say not, since a Google search for "tour" only turns up public transport and biking tips. The hostel might be a good place to meet up with like-minded people, once you're there.

https://www.danhostel.dk/en/news/guide-mon-denmarks-only-five-star-attraction

https://www.visitcopenhagen.com/copenhagen/mon-island-and-cliffs-gdk450090

https://www.rejseplanen.dk/webapp/index.html?language=en_EN

I don’t know anything about Tour companies (not something I use as a local) but I Think Laura might be right.
Møns Klint is a bit out in the sticks, but it is still doable by public transport. Type in Geocenter Møns Klint as your destination in the travel planner www.rejseplanen.dk. I suspect there will be better bus connection in the summer, but the time tables have not come out yet.
It is more practical to get there by car though, but it might be a bit pricey to rent a car on your own.
when you decide the way you do it, I would be happy to suggest interesting stops along the way.

Thanks Laura and Morten! I was hoping for better news, but that still won't stop me from visiting the cliffs.

I plan to go there on Saturday or Sunday, 9-10 of June. I would indeed appreciate some suggestions for places to stop at on the way. Any tips on activities, where to eat and what to bring would be appreciated as well. Would a swim be possible there or should I save it to the beach?

The main point of entry to the cliffs is the GeoCenter http://www.moensklint.dk/uk?lang=en
It has exhibitions, activities and a cafeteria. The beach there is not ideal for swimming, it's mostly rocky and June might also be too cold.
If you do the trip by public transport, you need to take the train to Vordingborg and change to a bus there. Vordingborg is home to the ruins of one of the largest medieval castles in Denmark, which is worth a look. There is also an exhibition centre https://www.danmarksborgcenter.dk/en.
If you rent a car, that opens up a whole range of possibilities, depending on your interests.

I will be in Copenhagen this August with kids (7 and 9) and plan to rent a car for a day to visit moms Klint. What stops do you recommend along the way?

The castle museum in Vordingborg could be a good sight for kids, but keep in mind that it is almost a 2 hour drive to Møns Klint Copenhagen, so maybe you don't want to pack in too many sights in one day.
Not far from the cliffs you could make a short stop at either Elmelunde church https://en.wikipedia.org/wiki/Elmelunde_Church or Keldby church https://en.wikipedia.org/wiki/Keldby_Church to look at the medieval frescos. Back when the church service was in latin and the ordinary peasant was illiterate, these frescos where like a cartoon version of the bible. The massacre of the innocents is particularly gory.