No. We won't. It's a constitutional impossibility for the American government to 'fall' outside of a revolution. The only thing that will happen is that two years from now there will be elections but not for president. I consider the fact that the government can't fall to be a bug in the US political system but it's not even the biggest one.
For all the reverence the 'founding fathers' get they could have done a much better job setting things up. America got rid of royalty, then brought it back in by electing them, and not in a ceremonial function either. It's a huge problem (that, and the 'two party system').
Technically the two party system could change but things are set up in such a way that no third party will ever gather enough votes even if you were to say pit two un-electables against each other.
It's like people don't remember that there was a president that called a press conference, spoke to the people, put on a jacket, got in a helicopter, and flew away from the White House forever.
Yes, and other people seem to forget that that did not lead to new elections.
Presidential elections happen on schedule. If Trump should leave that gives you Pence, if Pence should leave that will give you Ryan. And likely by that time it will be 2020 and you'll have those new elections but until then you'll be stuck with a Republican as president, like it or not.