Your first sentence is not grammatical English, so it will not sound natural to any native speaker. The reason is that 'self' is the indirect object of first supporter, but your sentence doesn't indicate the relationship. You would have to say
You are your own first supporter
Your second sentence fixes this problem by explaining that you are the first supporter of yourself. There is functionally no difference between 'your self' and 'yourself' in this case. They mean the same thing.
But neither sentence can be used by itself without sounding strange, because the phrase 'first supporter' is not a common phrase in native English. It's confusing because we don't know what is meant by the word 'first'. Does it mean you are the earliest one, the one that comes before the others sequentially? Or does it mean you're the most important one of all your supporters?
Also, the word 'supporter' is not usually used in a general way. It needs a context. A person might have supporters at work, or political supporters, but not just 'supporters'.
Your question says that you want to say a person 'should be' the first supporter of their self, but your suggestions use the verb 'are'. 'Are' doesn't mean the same as 'should be'. If you want to say that someone 'should be' their own supporter then use an imperative word like 'be'.
Be your own first supporter
The idiomatic phrase that comes to my mind for saying the same thing is
Be your own best friend
You could also say
Support yourself first
This could be understood in several ways, depending on what the word 'first' is referring to. You would have to clarify what 'first' means by adding context.
support yourself before you support others
support yourself before you do something else
support yourself before others support you