Insurrectionists cannot be a member of Congress.
The 14th Amendment, Section 3 states:
A movement is afoot to challenge Madison Cawthorn from running for re-election because of this. They hope to expand the movement to challenge all the other members of Congress who participated in the insurrection.
The challenges for them will be (a) defining exactly what behaviors constitute insurrection for purposes of this, and then (b) figuring out just what degree of proof is needed to nix a candidate, and then (c) proving that.
In the case of North Carolina, it apparently doesn’t necessarily take a conviction of insurrection in a court of law to deep-six a candidate. According to the NYT article, it requires the challenger to establish some sort of significant suspicion, at which point the burden of proof shifts to the candidate to show that he did not engage in insurrection.
ETA: If you find the above NYT article paywalled, you can also find a summary of it at Raw Story: