Hi @Raspada,
Thanks for sharing your question with us!
Naturally, the Klaviyo extension for WooCommerce handles adding .js script to the checkout page, which would allow forms to publish on the page. However, there can be issues if you are using a 3rd party plugin app for checkouts which may be preventing the script from being added automatically or your klaviyo plugin is out of date. I would first ensure that these two possibilities aren’t the issue before you insert the code to your checkout page.
However, if these two possibilities aren’t the issue, then yes, I would recommend adding Klaviyo.js to the checkout page for the form to work as it doesn’t seem to currently have the form snippet within its HTML. I'd recommend going to your form building and clicking on the "Install Code Snippet" button to access the snippet. Then you'll need to add this snippet to your checkout file within your Woocommerce account.
Thanks for participating in the Community!
-Taylor