That is called high-level synthesis and there are some tools for this available (and I know at least one company who produces a proprietary toolchain - including hardware - for this).

In my experience, high-level synthesis makes things a lot easier for the programmer, but you still have to be aware of very low-level details (and you still can get bitten by abstractions that you don't fully understand).

That company might fruitfully make their system available within a datacenter / "VPS"-type context, eg the way Cloud9 does it. (https://c9.io/, you make a free account, you instantly get a Web based editor connected to a Debian VM)

If their system accepts "mostly ordinary" code in eg C, a fair amount of stuff would probably work with it, so it could scale beyond educational exploration/tinkering, too.

Here is a service I know of: https://reconfigure.io/

I think they did exactly that with Amazon AWS instances (although I have to admit that I don't know the specifics).

Like this?

https://www.mathworks.com/solutions/fpga-design.html

There are plenty of C-to-HDL languages (https://stackoverflow.com/questions/8988629/can-you-program-...) and there's some interesting tidbits about them in this SO link.