Vayama Travel Website- Is it legit?

I've never heard of them, can't tell you anything about them. That said, I never buy tickets from places like that, as there are just too many ways for something to go wrong, and when it does, who ya gonna call? I'm guessing that "Vayama" doesn't answer phones and has no ability to fix things. I prefer to deal directly with the airline I'm flying on.

They do not have 'live' pricing. Airlines do not give away cheap airfares to resellers. These fares are just to attract your attention. Book with an airline.

Yes, I’ve purchased from them. I didn’t think twice. But why in the world are you buying one way tickets?

I’ve used Vayama a few times and never had a problem with any purchase. If I’m not mistaken they are primarily a Europe centered operation based in Amsterdam.

Some reading for you

https://www.bbb.org/us/ca/san-jose/profile/travel-agency/vayama-1216-232273/complaints

I've used Vayama once for a Virgin Atlantic flight to Miami. It was cheap, the code was for the cheapest flight ticket with no changes etc. We did encounter some problems with schedule changes but this was ultimately resolved on the phone.

Would I use them again? Most likely not, I much prefer dealing with the airlines direct now.

We are flying into Zurich and out of Rome so looking to book each leg and came across the Vayama website through Kayak.

You are planning a textbook "multi-destination" itinerary. Use a multi-destination search function to buy the entire trip. These routes are not a pair of one-ways. I suggest looking at matrix.itasoftware.com to see the possibilities. Matrix is run by Google but does not sell tickets. You can take the information and go to the airline website, preferably the trans-Atlantic service. It will likely book the entire trip. Vayama has been around a long time and there are plenty of others. Difficulties arise beyond the basic ticket, for such things as advance seat assignment. It's easier to buy directly from an airline, and the chance of a discount through an on-line travel agency is very small. I only use such agencies to book a a trip combining several airlines that do not sell each other's tickets.

You can find hundreds of comments about Vavama on the TripAdvisor Air Travel forum. Here is one example:

https://www.tripadvisor.com/ShowTopic-g1-i10702-k9250578-Vayama_Scam_or_just_Angry_Reviewers-Air_Travel.html

You are trying to book one-way and those tickets are much more expensive than a round-trip with most (but not all) airlines. What Vayama apparently does to get you the lower price is book a roundtrip ticket starting with the outbound trip on the date you request, and a random return date which younwill not use. At least that is what I have read as an explanation for their lower prices on one-ways.

The best advice is not to use third parties for booking trips like this, you should be able to get decent prices using the "multi-city " function on any airline's website. Use Google flights to identify potential candidates and then go to the airline website to book.

I used Vayama only once. I was able to travel on the tickets but they do not use live pricing from the airlines. They will sometimes jack up the price once you actually try to book, to match what the tickets actually cost.
When I booked with them I had to speak multiple times with people in a call center that was clearly overseas. The online "booking" I made online wasn't actually a booking, they said, just a request for a booking. They increased the price - not so much so that I cancelled the transaction - but I was annoyed. They claimed that the airline increased the price between the time I made the "booking" online and the time I spoke with them on the phone.
The whole thing felt a little dicey so I haven't used them since.
You usually get what you pay for and if it seems to good to be true...well. Ya.
$3,000 one-way from New York to Zurich sounds a bit high unless you are going in a class above economy, even for two people. I have flown there from LA for quite a bit less.
Just to add my two-cents to the comments above, I have found that flying one-way can actually be cheaper than round trip in quite a few cases. Not always, but I fly to Europe and back several times a year and one-way can be a downright bargain sometimes.