Using iframes as you described should work just fine in pure HTML.
You can also use a static site generator program: it can take as input one file for your navigation bar, one file for each of your pages, and output your pages with the navigation added. If you use a static site generator you would re-run it every time you change a page or change your navigation, then upload the outputted files to your webhost. Some popular static site generators are Jekyll, Eleventy, and Hugo, and there are more marked with “SSG” in the SSG section of the resources list.