I was nearly in this position in 2011 (IIRC). I dreaded the idea of calling my boss while on vacation and giving her that news. I ultimately decided not to take the new job (due to other considerations, not due to the potential awkwardness of the phone call I was going to have to make). But that's what I would suggest: Call your manager, today, and give them your notice.
I wish I could do this in person/I'm sorry this comes on such short notice/I understand the timing here is less than ideal/etc, but I wanted to give you the earliest possible notice that I've decided to pursue another opportunity. Please consider this my notice, and my last day will be October 11.
Depending on your tenure/value to your current employer, they may attempt to counter-offer, so be prepared for that eventuality. What will you do if they offer to match the new salary/benefit/etc.?
My last offer, employer at the time asked if they could counter, and I told them "respectfully, no." Primarily because it was the right time to move and it was a good opportunity, but also because the salary differential was so great that I don't think they could've matched it, and I would not have been comfortable with the arrangement for a few reasons if they had.
do I go in early and inform them that I will be leaving the company my start date has to be before October 14. Or do I go in on 3 October and say hey I got a new job I have to leave before the 14th. I’ve never quit while on vacation not sure if that is respectful and courteous to my employer but I’d rather tell them now then later when I come back because either way I’m leaving
Telling them now maximizes their advance notice. I would consider this more courteous than telling them when you return, although a difference of a few days seems pretty negligible. Since you seem to have made up your mind, there's no reason not to tell them right now.
It may seem/feel a bit untoward to deliver this news while you're on leave. So what? You have to balance that against the timeliness of the notice. There's no perfect solution. If you were not on leave, would you tell them today? Or would you wait until the 3rd?
I guess I'm having a hard time understanding why (aside from the awkwardness of quitting while on leave) the fact that you're on leave is a serious consideration here. They expect (formally, or probably informally at least here in the US) two weeks advance notice. Give them that.