JANESVILLE, Wis. — When dozens of high school students from across Wisconsin marched through his hometown, chanting his name and demanding new gun control laws, Paul D. Ryan, the House speaker, was not around to hear them, having just wrapped up a trip to the Czech Republic.

But Randy Bryce was.

Mr. Bryce, for those who don’t watch MSNBC, is better known by his Twitter handle, “Iron Stache” — a nod to his occupation (ironworker) and his thick horseshoe mustache. A Democrat, he has become a liberal media darling of sorts, as he seeks to do the unthinkable: unseat Mr. Ryan in Wisconsin’s First Congressional District this fall.

But Mr. Bryce is not the only candidate hoping to dethrone a congressional king. Another Democrat, a schoolteacher named Cathy Myers, is also running for her party’s nomination. She is irked that national Democrats and progressives — including Senator Bernie Sanders, the former presidential candidate — have thrown their weight behind Mr. Bryce.

And on the right, Republicans are confronting an embarrassing spectacle: A white nationalist and anti-Semite, Paul Nehlen, who lost to Mr. Ryan by 68 points in the 2016 Republican primary, is running again, this time flaunting his bigotry to gain a national following. His Twitter account was suspended in February after he used it to make racist comments about Meghan Markle, the fiancée of Prince Harry.