A third party would need more than a couple issues to drive it. That type of party wouldn't know what to do with the Presidency even if they obtained it.
That actually happened in the 1990 Ontario provincial election. The New Democratic Party (our third major party) typically gets a small share of seats, and generally represents the left in our federal and provincial governments. They're a long-established party, but they have a lot in common with the fringe parties, in that they seldom win enough to have any real influence. Usually, the most they've been able to do is hold the balance of power by either supporting somebody else's minority government or forming a coalition with the runner-up.
I should explain for the benefit of Americans. We in Canada, as in other Commonwealth countries, use a parliamentary system. We don't elect our leader independently of our representatives. We basically have parliament at the federal level and legislatures in each province that all work more or less the same. Each riding (electoral district) elects a member. Parliament has a seat for each riding. The party winning the most seats forms the government, with their leader becoming prime minister (or premier in a provincial election). The heads of the various ministries are then appointed from the elected members of the winning party. The party in second place becomes the official opposition, and appoints a "critic" for each of those ministries.
However, when a party fails to secure a majority of seats, and simply has more than anyone else, other things can happen. In order to pass legislation, the party in power needs the support of one or more of the other parties. That's when a little party like the NDP can hold the balance of power, as they've done a number of times. Minority governments are fairly cautious, because they are vulnerable to no-confidence votes. Basically, if certain important pieces of legislation, such as a budget, fail to pass, a new election is called. That happened to our federal government a couple of years ago. Federally, there is actually a fourth party with considerable influence, the Bloc Quebecois, who do extremely well in Quebec, but don't run candidates anywhere else. But that's really beside the point.
A party failing to secure a solid majority might also never get the chance to form a government if two or more of the other parties agree to form a coalition to secure a majority. They basically combine their seats and form a government. The NDP has also done this. I remember them joining with the Liberals in the late 80s to take the Ontario government from the Conservatives.
Anyway, where was I? Oh yes, the Ontario NDP, most idealistic of parties, chock full of academics and perhaps a little short of business people, went into the 1990 with the usual aim of getting the message out and winning as many seats as possible. Their best outcome would be a minority government that they could influence. However, voters were p**sed off with the current Liberal government, and they were still p**sed off with the previous Conservative government. So, a lot of voters said "To hell with it, I'm voting NDP."
I don't know if all of those voters actually wanted to try the NDP, or if maybe some of them thought of it as a protest vote with the chances rather slim that the New Democrats would actually win. They did, with a solid majority. Nobody was more surprised than the NDP themselves, who never expected that they'd actually have to form a government, and might not necessarily have had the best-equipped people in the legislature for the job.
Five years later, with everything in a mess, the voters booted the NDP out in favour of an ultra-conservative Conservative government that took things in the opposite direction entirely.
The NDP does get elected in other provinces, but that was the only time the Ontario NDP got the chance. They've never won federally, and I doubt they ever will. Most people consider their strength to be pestering the governing party, not actually governing.