There's a contradiction in what you want, viz
"I would like any reader who visits our blog to be able to request
By definition, once they visit your blog, they already have access.
But I'm guessing that what you really want is that
"I would like any reader who tries to visit our blog to either get
into the blog, or to be invited to request access."
Blogger does not have any tools to faciliate this.
But, I wonder how potential visitors who don't have access yet would find your blog? It won't be via search (because private blogs aren't indexed). So it must be because someone has told them about it.
If it's you who is doing the telling, then you could set up a 2nd public (but possibly not indexed) blog, which just has one post saying "Click here to see our blog - or email [email protected] to ask for access" - and only ever share the address of this public blog.
Or if it's your readers who might tell their friends, then put a widget etc into the blog saying "use this to share myBlog with your friends) - and again, have it link to the public "front window".
One warning though: Blogger allows only a maximum of 100 invitations to a private blog, ever. Once you've invited 100 people, you cannot invite any more. If there's a chance you'll go over that limit, then you might need to use a different blogging platform.