WASHINGTON—With polls showing Hillary Clinton and Donald Trump entering the general election as their respective party’s most unpopular nominee in recent memory, third-party candidates see a once-in-a-generation opportunity to play a significant role in the fall presidential campaign.

Buoyed by polling showing a renewed appetite from voters for options beyond the Republican and Democratic parties, Libertarian and Green candidates are vying to participate in the debates, inject their issues into the political conversation and bring their message to a wider audience.

In addition to the Libertarian and Green Party challengers, a handful of disaffected Republicans are looking to recruit an independent conservative candidate to challenge Mr. Trump. Former New York City mayor Michael Bloomberg also flirted with an independent bid before determining he had no path to victory.

Bob Shrum, a veteran Democratic consultant who has worked on a number of presidential campaigns, expressed skepticism that significant numbers of voters would flock to a third-party candidate, but acknowledged that in a close race, even a few percentage points could potentially be disruptive. “I think they could take some votes,” he said. “It depends on how close the race.”

The Libertarian Party this weekend at an Orlando convention will have a chance to pick the most experienced ticket in the party’s 44-year history. Two-term New Mexico Gov. Gary Johnson is vying for the presidential nomination, while fellow Republican and former Massachusetts Gov. William Weld has declared his intention to seek the vice-presidential spot.