My issue is that there is no "subscribe via email" function to this page, only a subscribe via RSS.

Not everyone has an RSS reader.

Ideally I'd like a "subscribe via email" option on the blog page.

Currently I can't see a way of adding this.

If I was to set up a wordpress blog and then integrate that, would people be able to subscribe via email that way? Would they do this by visiting my wordpress page, or would it be integrated into the blog page of my website?

I have already set up a campaign for RSS, but without a way of capturing subscribers, I can't actually create a mailing list.