I’d google vegan restaurants, NY then take a look on Tripadivisor at the reviews of a couple. I’m a pescatarian and finding non meat options isn’t that hard. Food isn’t particularly cheap but the portions are huge. Various of the food trucks you’ll find across the city have vegan options (although I’d imagine they’d be cooked on the same equipment as meat so....).
As first glance, NYC doesn’t seem to have that many free/cheaper options but they do exist! There is an NY attractions pass you can buy which will get you into a number of the bigger tourist attractions - we didn’t get this so don’t know the cost. Alternatively you can book various of the attractions online ahead and spread the cost over the next couple of weeks.
We got show tickets through TodayTix, it’s an app I use a lot for West End Shows and has numerous NY discounted show options where you can select your own date, seat etc. Just download it and change the location to NY - their reps meet you outside the theatre with your tickets.
Taking the ferry over to Staten Island is free and gives you a great view of the Statue of Liberty. The High Line (converted over head rail line) is also free and is a great wander on a sunny afternoon with lots of benches and art by some pretty well known artists. MoMa (museum of modern art) won’t break the bank & houses one of the best modern art collections I’ve ever been into. Walk the Brooklyn Bridge, Central Park etc etc.
One easy (but more hard work!) option re eating and saving on costs would be to rent a studio apartment and partially cook yourselves. We rented one in Brooklyn which was in a lovely old brownstone and was cheaper than many not so great looking Manhattan hotels. There are many legal restrictions on renting apartments in NY (it’s estimated 45% of AirBnBs in NY are illegal) so you’d have to look carefully at this as an option). The owner & family lived in the rest of the building we stayed in, separated from our lovely & very tasteful studio by an MDF door making it a legal let (booked through HomeAway who’s legal team I spoke to) but this is a bit of a grey area with changing legislation so if this is something you look at to save on costs, it isn’t without a small element of risk and do your homework. Quite a few other Mumsnetters have done it though and it was definitely the cheaper and nicer option!