Hi,
The Recaptcha v2 plugin we provide for HikaShop works fine even without cookies.
I've just tested myself on a test website along with the "EU e-Privacy Directive" extension which allows you to decline/accept cookies on the website just to make sure...
Even with the cookies declined, I was able to register on the website through the HikaShop registration form while validating the catpcha.
In order to be able to have a product, and thus access the checkout, cookies need to be activated. It can't work without and that's the same with the vast majority of ecommerce systems. So you wouldn't able to access the checkout of HikaShop (and the registration form there with the captcha there) with the cookies refuses.
And even having the registration form working without cookies is of little use as you wouldn't even be able to log on the website without cookies.
To be able to have persistent information from one page to another on a website requires a user session, and to know which user session is linked to who, a user session token needs to be stored as a cookie so that the browser can send it with each page so that the server can recognize the user and get back the persistent information (like the logged in state, the cart of the user, etc).
So an ecommerce website were you would decline the cookies wouldn't allow you to do much regardless of whether the catpcha plugin supports working with or without cookies (and it works without them), or whether you use HikaShop or another ecommerce extension.