According to the OP’s strict definition of “normal” (people without connections and access to big money), I’d say No, they do not have any chance of being elected president, at least in the US.
But, I would not write off the possibility of less tightly defined “normal” person – namely, a “non-politician everyman-type” individual – getting elevated into the presidency in a single shot, slim though that possibility may be. It would take a perfect-storm-style mix of circumstances.
One of the essential ingredients already exists: political parties. They possess the machinery to conduct a national campaign for a ham sandwich, if they chose to use it. In the smoky good ol’ days, one or more kingmaker party bosses could pick a nominee that the delegates would generally rubber-stamp into the candidacy. These days with primaries and all, it’s harder to pluck dark horses out of the shadows and onto the ticket, but with ingredient #2 it might be possible to get the party forces in line behind Candidate Schmo before there are too many pesky challengers in the field.
Now, it would probably help Candidate Schmo’s nomination chances if his party is weak and desperate, like a losing football team needing to throw a Hail Mary pass to win the game. Of course, once he’s on the ticket, it helps him if the party is strong, so it can get him elected. This part is, admittedly, tricky.
Essential ingredient #2 is an overnight hero (our Mr. Schmo) without any prior political of financial assets emerging from some gigantic unforeseen crisis. This has never happened on the national level, though there have been some examples that sort of come close: Rudy Giuliani (during 9/11), and the military leaders Andrew Jackson, Teddy Roosevelt and Dwight Eisenhower (leading armed troops to victory during wartime). None of those guys were nobodies before they took their bows on the national stage, of course, but they could easily have faded without a trace without their big, defining moments of glory.
I would argue that, with today’s acceptance of instant celebrity and the pervasiveness of the mass media, a Mr. (or Ms.) Schmo could win the nation’s heart with one grand act of bravery, generosity or quick-wittedness. If Mr. Schmo happens to be blessed with natural Clintonian charisma and talents, that ups his odds considerably. If one of the parties hoisted Mr. Schmo on its shoulders, he might have a shot at the presidency.
Now, that leaves essential ingredient #3: a weak, worn-out, mismanaged or imploding opposition party. Beating Candidate Schmo should be an easy task, but nothing is a certainty in politics. Maybe the people have had enough of them. Maybe they pick a weak opponent. Maybe those naked photos of Mr. Opposition sleeping with Cub Scout Troop 72 finally surface two weeks before Election Day. Anything can happen.
So, IMO, a Mr. Schmo can get elected president. Likely? No. Possible? Yes.