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In short, yes, you can sue, but you almost certainly would not win. Anyone can sue anyone for anything. A lawsuit is merely a legal complaint. It does not require that the person be right when the suit is filed. That is what the judges are for: making determinations of law. You can literally sue God or XXXXX XXXXX or Harry Potter. There would be "procedural" issues in "serving" them with legal notice, and lack of service would allow a dismissal of the case. But you could still "sue" them. And you could sue the Tea Party, Republicans in general, Congressmen, but you would run into procedural issues, as well as substantive ones that would almost certainly mean that you would lose your case.
First, there's the issue of "standing": You have to have an "injury in fact" to be able to sue.
That means that you, personally, have to prove how you have been harmed by another's actions. If your neighbor gets into an accident with a third party, you can't sue that third party on behalf of your neighbor. Only your neighbor could sue. And if someone almost hits you, but doesn't, even though there was a close call, you would not have any "injury" to be able to sue. So to be able to survive a motion to dismiss, you would need to be able to show how you, personally, are affected by this.
But the bigger issues is the issues pertaining to "immunity" and "proximate cause". That is, you can't successfully sue Congressmen for official actions, as there is a "congressional immunity" for these actions. Because they make the law, the government (and governing members) can give themselves immunity, and you can't successfully sue them for doing something or not doing something (so long as they have that discretion).
As for the Tea Party and Republican members, you would need to first show that they have a legal duty in the first place to avoid default, which there is no legal precedent and they personally have not signed any contract or debt instrument obligating themselves or America in general to this debt, so there would be no duty under that. It's possible (although highly highly unlikely) that a court could find an equitable duty.
But then there's the issue of "proximate cause":
To be able to hold someone responsible for their actions, they have to be both the "but for" and "foreseeable" cause of the incident(s). For instance, suppose John Doe orders a pizza from Dominos. A Dominos driver, in trying to get that pizza to John Doe in 30 minutes or less, runs into you negligently. While you can successfully sue Dominos and the Dominos driver, you couldn't successfully sue John Doe. Even though John Doe is a "but for" cause of your injury (in that had he not ordered a pizza, the Dominos driver would not have run into you) he's not a "proximate" cause of your injury, because the true cause (the Dominos driver) can be seen.
In the same sense, even if the Republican party and Tea Party cause Congressional members to vote for default, it's the actual Congressional members that do so (like the Dominos driver) and hence are the proximate causes.
And because they have Congressional immunity, you could not successfully sue them.
In short, the answer to your specific question is "yes", you can sue, because you can sue anyone for anything, but to answer your implicit question of whether you can win such a lawsuit, no, you could not for the reasons listed above.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for my answer unless and until you rate it a 3, 4, 5 (good or better). Thank you, XXXXX XXXXX luck to you!