The simple answer is no – search engine submission isn’t necessary. But that doesn’t mean you can’t do it.
Every search engine today, whether it’s Google, Yahoo, Bing, Yandex, or any other, can crawl and index pages by following links. A single inbound link from any already-indexed page will identify your page to the engine and if that page links to other pages within your site, they will also be indexed… and so on. And if you’ve got a very popular website, Google (especially) is likely to crawl and index your content within a few minutes.
That’s not meant to diminish the need for backlinks. (In fact, acquiring back-links may be the most important of SEO.) In theory though, a website owner shouldn’t have to ’scout’ or ‘hunt’ for links. If the content you’ve written is of interest, useful, and/or important, there is a natural tendency among web users to link to information. How do you know if it’s well written or of interest? Test it out on social media. If you get some shares or likes right away, you’ve probably got something interesting.
That said, submitting your site to search engines that allow you do to so can’t hurt. If you’re the impatient type, Google has a tool you can use to submit your content, albeit one URL at a time. You can find Google’s tool in Google Search Console. (If you’re using Bing, and you’re logged in, you can use the URL submission tool in Bing Webmaster Tools.) Click Google’s URL inspector in the lefthand navigation, paste the URL you want to add into the box at the top of the next page, and press Enter. Google will check its index to see if the URL is present. If it’s not in the index, Google will present you with the opportunity to request indexing. Don’t request indexing more than one time. It doesn’t help or get your site crawled any quicker or more often.
Submitting your site to most search engines doesn’t cost a fee. There are exceptions, and these are usually industry-specific search engines.