Years ago, the answer was yes. Indeed, there continues to be a perpetuating myth that search engine submissions are still required. However, the search engine companies are pretty clever and are constantly evolving which means that they do NOT need to be directly notified when a new website, or page on a website, is created. They do it automatically.
Nowadays, most search engines crawl the web regularly, looking for websites to index. They start at one website with a large number of outbound links (this is usually a directory) and then they follow every single link that they come across, indexing each page that they arrive at. Once a page has been indexed, they follow the links from that page. This goes on and on until they simply run out of links to follow. In order to keep this daisy chain of links going, inbound linking (links found elsewhere on the internet that direct users to your website only) is very important.
So, unless you have literally just gone live with your website, it’s highly likely that if a search is done, it can be found (something which you can obviously easily test yourself)!
Where some confusion arises is around the issue of a website not showing up for the keywords that someone wants to show up. However, this has absolutely nothing to do with search engine submissions. You can’t just contact Google or Yahoo and tell them that you want to be ranked for certain keywords…it just doesn’t work like that! Which keywords your website ultimately gets ranked for is dependent on search engine algorithms. So, in theory, a website owner shouldn’t have to ’scout’ or ‘hunt’ for links. If the presented content is of interest, useful, and/or important, there is a natural tendency among web users to link to information. This is the basis for the Google PageRank algorithm.
Having said all of this, submitting your site to the search engines can’t hurt and you only need to do it once. It doesn’t cost a fee to submit your site to most of the search engines and it only entails about 5 minutes of added work assuming you only submit to the big guys. Many SEO companies assert that you need to pay to be indexed but this is certainly not the case so steer clear of doing this.