Actually the Republican party and the Republican establishment do not want Trump.
So? The party (and the resulting government) is more than one person.. though with the amount of party line voting that goes on its sometimes hard to remember that.
Electronic media including emails is considered greatly inferior.
I think you need to join the 21st century. Email is no longer the unloved stepchild of communications. Most people treat it as a legitimate form of communications (and many companies don't even bother reading paper letters/resumes/etc anymore.. which I know isn't the same as the government but it definitely indicates a broader trend.)
And still even a letter is vastly inferior to a vote
No, no it isn't. For exactly the reason I mentioned: One vote gives you 1/350millionth of a say in government, applied to a broad range of issues, once every 4 years. Letters can be written whenever you want and can be specific to an issue that's important to you -- and you can even send it if the elected official isn't the one you voted for!
Trouble with letters isn't that they are useless. The trouble is that people assume if they send a letter that the politician will magically agree with everything they say and implement their ideas immediately. Which is not how things work. Just because you want something and even just because you spent the time drafting a letter, doesn't mean they'll listen to you. But then again they might so its always worth a try if your opinion is that strong.
Again, digital, largely considered to be of little value.
20 years ago perhaps. Maybe even 10 years ago. But that's no longer the case. People have realized that just because I use a keyboard instead of a pen doesn't make my thoughts less relevant, and similarly with online petitions -- just because the "signatures" are in the form of email addresses doesn't mean they aren't just as valid as a written one.
And the online petitions have scale on their side. If you go out canvassing you might get a few hundred signatures. That might be enough to get your local city government to install a new stop light but spread across the hundreds of millions of people in the entire country, 100 signatures from one tiny little area isn't even worth looking at.
100,000 "signatures" with representation from every state on the union on the other hand.. that's hard to ignore. Even if you allow a little fudge room for fakes and duplicates, that's a hell of a lot of people as I said. Even if the politicians ignore it, the media sure as hell won't and once they get involved politics kind of has to keep up.
Of course just like a real petition, you can't just collect signatures and sit on them. There still has to be someone running the operation who's willing and able to take the results to the politicians and/or media. But I'd just assumed that was obvious.
Look at the White House's online petition system
I'd rather not. The whitehouse running their own petition system would be like letting a criminal be the judge at his own trial -- you'd never get a guilty verdict.
Groups like the EFF and OpenMedia.ca as well as many similar groups internationally.. those are the ones you want to go to. I of course tend to focus on internet issues but you can find organizations setting up these kind of petitions for environmental issues and I'm sure many others.
Just keep in mind that the petition is only one step along the way. Someone has to first raise the issue, then raise awareness of the issue, then start and operate the petition, then take the results to someone who will listen and has the power to do something about it. Just like a door-to-door petition process. The petition itself is only one step along the way but its the most important one since its adding the voice of the people to the issue. In recent years, these online petitions have really been the ONLY voice against the continual lobbyist pressure for things like the TPP.
And that's fine. In fact its good -- its the first time in history that a large portion of the voting population have a voice outside of election time, even if its filtered through someone's petition system. And it DOES make real change happen. As long as you recognize the fact that on occasion, it won't be enough and your favorite issue will still go in a direction you don't like. That's life. But it gives us the chance to try, for a much much wider definition of "us."