We should be able to add customers without any interference from Jetpack.
The Register button disabled as soon as I started typing a password.

I've left this up on my site if you'd like to see it in action. It's a live site, but not in use at the moment.
https://deerdash.com/my-account/
Internal reference: 606355-zen
cc @pmciano
Update: this bug is happening when any Jetpack module is activated (not just the Extra Sidebar Widgets module).
This is not a Jetpack issue. To start with, I deactivated Jetpack and tested. Same issue happened.
I checked it and found that it's related to pass length. Looked up the scripts loaded in the page and saw there's the password strength meter from core and one that uses it in WooCommerce. The relevant code is
https://github.com/woocommerce/woocommerce/blob/master/assets/js/frontend/password-strength-meter.js#L32
Most helpful comment
This is not a Jetpack issue. To start with, I deactivated Jetpack and tested. Same issue happened.
I checked it and found that it's related to pass length. Looked up the scripts loaded in the page and saw there's the password strength meter from core and one that uses it in WooCommerce. The relevant code is
https://github.com/woocommerce/woocommerce/blob/master/assets/js/frontend/password-strength-meter.js#L32