Yes, a single page website can have a blog, but it's not mandatory, it depends on your main objective, if your website is already performing well, and you're receiving inquiries or sales without needing a blog, it might not be a priority, but if your intention is to attract organic traffic from google or position yourself as a leader in your field, having a blog, even a simple one, can be a game changer for your project.
The technical aspect is simpler than it seems, the most common and recommended approach is to have the blog in a subfolder like site.com/blog, with a link in the main menu that leads to that section, the articles don't need to be integrated into the homepage itself, this maintains the simplicity of your website while adding the SEO power that a blog can provide, you can target long tail keywords and answer specific questions your audience has.
The decision depends on your growth strategy, if you need to be found through searches related to your business, a blog is an ideal tool, if your business relies on referrals or direct traffic, it might not be urgent, there's no rule against it, so if you think it will benefit you, go for it, the blog doesn't compete with your one page website, it complements it.