In short, no.
If you have read our article on SEO Basics for Tutors, you will know that SEO is important to helping potential students find out about you.
If your website doesn't use good SEO principles and practices, it's likely that potential students can't find you. On the other hand if you do work on and improve SEO for your website, then more and more students will find you and contact you. So SEO is definitely important to work on.
You will notice that a lot of companies and professionals offer SEO services at a cost. They usually make claims such as 'rank #1 in Google'. At first glance it sounds like a good idea to pay somebody else to take care of SEO for you so you can focus on your tutoring.
In reality, it's not a good decision. Countless companies or 'experts' you see listed online can be considered scammers. They'll take your money and you won't see any real results. Many others can be considered ripoffs who charge you a fortune then use shady practices which could actually get you penalized by Google.
There are companies that are legitimate and do a good job, but the problem is that it's extremely difficult to tell the scammers or shady practice people apart from the real deal. We recently had one tutor who almost payed over $300 to a company we identified as one who uses shady practices. So this is a very real risk.
The truth is that it's not that hard to learn the basics of SEO yourself and do a good job on your own with a bit of effort. The reason this is the approach we recommend is because it's a skill not many tutors have. So if you take the time to learn SEO and apply it to your website, then you have a clear advantage over your competitors. It's a skill that you can apply whenever you need to and achieve good results.
So the basic answer is: no you don't need to pay somebody else because you can achieve good results on your own. Plus, you avoid the very real risk of being ripped off.