I'm a skeptic of most superstitions about "Natural", I can understand eating things healthy, but floors need as hard of a surface as possible so they're not damaged and today's water based finishes are so safe you should not be worried. Just stay away from the oil based ones or anything that uses a solvent or isn't a 2 part catalyzed system. The cross linking catalyst in the 2 part systems is like epoxy 2 part systems, they make a chemical bond which freezes all the material in place so it never will rub or flake or evaporate etc...
Pretty much all 2 part (catalyzed) water borne finishes these days are what I would consider "green". But there are some (IMO) lesser performing finishes which are even more "green", but you will end up refinishing with any of these "natural" coatings within a few years. The 2 part water borne finishes already have low VOCs (volatile Organic Compounds). Once the finish is sealed, it's just that, sealed, nothing is coming out of it, you're not going to get cancer from walking on a sealed floor. You'd be better off wearing only natural leather shoes if you're worried about a floor.
The old fear was the off gas (solvent based, 1 part, or oil based) evaporating from oil based finishes. Today most finishes have a chemical (catalyst) which doesn't rely on evaporation for the curing. Once this process is done (a day at most), it's purpose is to be as solid (not gaseous) as possible, hence you're not going to get anything out of that finish.
Bona and Basic are two good brands. I used a Basic Coatings finish on my floor I did 5 years ago and it's held up great. Bona Kemi is probably the most famous brand, but a bit more expensive. Bona makes a bunch of more natural finishes, you should check them out, particularly "Natural"
Then you have all the "organic" type finishes, which are mostly a hard wax and oil, and don't really hold up, they're expected to be redone relatively regularly (every few years).
Rubio Monocoat is a very natural finish, I would never use it, but I'm a bit more skeptical about the "organic" / Green movement, so take that with a grain of salt.
Today's waterborne 2 part catalyzed finishes are about as green friendly as you get, the old oil based solvent based finishes are no longer used most of the time, and that is what you're probably heard bad rumors about.
If you're spending a good amount of money (which you are), get a solid 2 part water based catalyzed finish and you will be happy. I did it, and it smelled vaguely of amonia, but the smell lasted less than a day, and it's been hard as nails today. Hope this was helpful and not too snipey! Organic doesn't mean safer! There are plenty of natural things (radon, lead, mold etc...) that you should be more worried about than your floor finish.