There are several issues with using the reCaptcha.

When setting the keys the text does not match that of the registration (perhaps Google has changed?)

Public reCapcha key -> should be Site key
Private reCaptcha key -> should be Secret key

The text above the reCaptcha is incorrect and relates to the "image" spambot
(To prevent automated registrations the board requires you to enter a confirmation code. The code is displayed in the image you should see below. If you are visually impaired or cannot otherwise read this code please contact the Board Administrator.)

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Edit: Disregard this error
I am getting the following error messages

[phpBB Debug] PHP Warning: in file [ROOT]/vendor/google/recaptcha/src/ReCaptcha/RequestMethod/Post.php on line 68: file_get_contents(): https:// wrapper is disabled in the server configuration by allow_url_fopen=0
[phpBB Debug] PHP Warning: in file [ROOT]/vendor/google/recaptcha/src/ReCaptcha/RequestMethod/Post.php on line 68: file_get_contents(https://www.google.com/recaptcha/api/siteverify): failed to open stream: no suitable wrapper could be found