Can I make

A new chess board 

And apply a chess.com system?

The site doesn't allow access to any APIs to make one-off electronic boards usable.

So, what does apply a new chess.com system mean?

make new chess board and apply chess.com new chess board

Are you asking about coding your own board interface?

Yes

I want to make new chess board disign

You could use something like user style sheets on the client side for a different design but you won't be able to do anything else and nothing with the Public API