I used to follow a certain blogger - but found out, recently, that now his blog is blocked & only open to invited readers. I do not know the author and would like an invite.Lack of access to the fancy clubs and parties, in any city, is a disappointment to many - except the lucky members of the "AAA" list. Private blogs have the same exclusivity. Life is not fair.
How do I contact the author?
Private blogs, in general, are a niche feature.
Blogger has been asked for other private blog improvements, such as increasing membership limit, and has generally declined.
No, private blogs are not that common - and private blogs that need more than 100 members even less so.
More would be members, than private blog owners, want easy owner contact.
The obvious solution, to have a landing page for private blogs, where non members can request membership, comes mainly from non members.
Look in the forum topics, for instance. Find the suggestions "I need a way to contact the owner of a private blog, and request membership!", then "I publish a private blog, and need a way for non members to contact me!!". You'll find more of the former, than of the latter.
Private blog owners make the decision to be private, intentionally.
Many private blog owners (a niche, in themselves) do not care to have a feature that enables non members to harass them, requesting membership. This is similar to the notion that people who want ownership of a dormant blog (URL) should be able to request assignment of the URL.
Private blog owners are given respect, by Blogger.
If you publish an anonymous or private blog - and don't publish a contact point, intentionally - Blogger is not going to force you to provide a contact point. Blogger respects the privacy, of people who have chosen to publish blogs to a restricted reader audience.
Owners of private blogs are given the right, to determine their own destiny. If someone publishes a blog, with no contact point, Blogger isn't going to force them to allow themselves to be harassed.
You may find a contact, if you work hard enough - but Blogger won't help you.
There may be a way to contact the current owner - but the would be member must find the way. Blogger supports anonymity, and privacy - and they won't violate that right that the owner has.