Q: If a visitor to your blog can't leave a comment, what should you do?
A: Readers need to have third party cookies enabled in their browsers in order to leave a comment in the embedded comment form. To make it easier for your readers, you could choose one of the other types of comment form (full page or popup).
I searched around the web and found some interesting data on commenting with Blogger blogs...here are some of the notes:
I attempted a test comment on the blog - with some difficulty. The blog uses an embedded comment form, with authentication (Google / OpenID) required, and CAPTCHA ("word verification") screening.
The CAPTCHA being used by Blogger is a nasty accessory. You should maybe try logging out of Blogger as the owner, then see if you can publish a comment as a blog guest - and see for yourself if you think your readers will appreciate the challenge. It took me over a dozen tries, to get my test comment to publish - and I have experience in dealing with the CAPTCHA. Remember - as the blog owner, you never have to use a CAPTCHA when posting comments.
Another possibility is cookie / script filtering. The best known culprit is the browser "third party cookies" filter - though there are other browser settings - and there may be other filters on the computer.
[FAQ] Why can't people post comments on my blog?
This post is a few years old, but it makes a lot of sense to me:
Using Dynamic templates:
If you have one of Blogger's Dynamic templates you may not have the ability to allow commenting. So if you switch to Dynamic, don't be surprised if your commenting is gone.
This is a Blogger bug, that I believe is in the works.
We'll be posting more tidbits on COMMENTING when we come across the info.