Updating the core does not typically impact any content or templating. 2.2.6 is really old so it is possible you’ll run into some small issues.
It’s recommended to upgrade one minor version at a time until 2.4 - so first upgrade to 2.3.x, then 2.4.x, and from there you can go straight to 2.7. Also upgrade extras to ensure compatibility. It’s possible to use the UpgradeMODX extra to make upgrading quicker/easier.
It looks like the form was built with FormitBuilder, and that has its own way of adding hooks and markup into the form. I haven’t used that before, but there appears to be documentation for it here. Including an example for recaptcha, which might be quite old but could point you in the direction of how to add it.