As I said in some comments, suing Pence in this case was obviously silly, as Pence attorneys themselves actually argued (and the suit was dismissed for lack of standing):
A Justice Department lawyer representing Pence on Thursday urged [judge] Kernodle to dismiss the lawsuit saying they had sued the wrong person as they raised “a host of weighty legal issues about the manner in which the electoral votes for president are to be counted”. “The Senate and the House, not the Vice President, have legal interests that are sufficiently adverse to plaintiffs to ground a case or controversy,” Pence’s filing said.
In general, the federal courts have deemed non-justiciable "resolution of disputes between parties who are not truly adverse".
As it was suggested by Dan Scally in comments might happen, there was an (unopposed) motion to intervene in the suit... but it was from Trump's electors from Michigan. There was actually one very late motion to intervene from a Biden elector from Colorado. I think there wasn't even any ruling on this motion before the lawsuit was dismissed. (Well, it was denied as moot, without specific reference to it, in the final judgment that dismissed the case [and thus all pending motions].)
Note that the plaintiffs filed a notice that they are appealing to the 5th circuit, so this isn't quite over yet.
As explained in Politico, the dismissal was mainly due to the uncertanty of the alleged injury:
A judge said Rep. Louie Gohmert lacked standing under a 1997 Supreme Court ruling that said individual lawmakers did not have standing to challenge a line-item veto law Congress passed a year earlier.
“Congressman Gohmert’s alleged injury requires a series of hypothetical—but by no means certain—events,” the judge wrote in his 13-page ruling issued Friday evening. “Plaintiffs presuppose what the Vice President will do on January 6, which electoral votes the Vice President will count or reject from contested states, whether a Representative and a Senator will object under Section 15 of the Electoral Count Act, how each member of the House and Senate will vote on any such objections, and how each state delegation in the House would potentially vote under the Twelfth Amendment absent a majority electoral vote.”
“All that makes Congressman Gohmert’s alleged injury far too uncertain to support standing under” the Constitution, Kernodle added.
Although they are filing an appeal, the plaintiffs don't seem too convinced it could succeed...
"Bottom line is, the court is saying, ‘we’re not going to touch this, you have no remedy,’" Gohmert said. "Basically, in effect, the ruling would be that you’ve got to go to the streets and be as violent as antifa and BLM."
The judgement actually detailed the issue of standing in re judicial intervention
in the actions of other branches of government, citing from the Raines v. Byrd extensively on that basically to show that
this is a strict standard. And after a couple of pages of that, it says:
[Gohmert] asserts that under the Electoral Count Act, “he will not be able
to vote as a Congressional Representative in accordance with the Twelfth
Amendment.” Docket No. 2 at 4 (emphasis added). Because Congressman Gohmert
is asserting an injury in his role as a Member of Congress rather than as an individual voter, Raines controls. [...]
Congressman Gohmert’s alleged injury is “a type of
institutional injury (the diminution of legislative power), which necessarily damages
all Members of Congress.” Id. Under these circumstances, the Supreme Court held
in Raines, a Member of Congress does not have “a sufficient ‘personal stake’” in the
dispute and lacks “a sufficiently concrete injury to have established Article III
standing.” Id. at 830.
The judgement then also cited Clapper v. Amnesty International USA
that the scenario laid out by Gohmert is too speculative to clear that bar:
Congressman Gohmert’s alleged injury requires a series of
hypothetical—but by no means certain—events. Plaintiffs presuppose what the Vice
President will do on January 6, which electoral votes the Vice President will count or
reject from contested states, whether a Representative and a Senator will object
under Section 15 of the Electoral Count Act, how each member of the House and
Senate will vote on any such objections, and how each state delegation in the House
would potentially vote under the Twelfth Amendment absent a majority electoral
vote. All that makes Congressman Gohmert’s alleged injury far too uncertain to
support standing under Article III. Id. at 414 (“We decline to abandon our usual
reluctance to endorse standing theories that rest on speculation about the decisions
of independent actors.”).
So basically it was a "double whammy" against Gohmert's claim (no personal standing (Raines) and scenario too speculative (Clapper)).
Trump electors' standing was basically denied because they didn't explicitly ask for Pence to count their votes as a remedy, only that he'd be given the latitude to do so. So again, they were speculating on his behavior.
That still leaves open that they can/may sue Pence after he does his thing (and if the lawsuit explicitly asks for Pence to act differently). Of some interest here, something like that did happen with Raines, as it was followed by
Clinton v. City of New York. In contrast to Clinton though, it would be much more difficult to show that Pence acts/acted illegally or unconstitutionally if he decides to open/"count" only Biden's electors' submission from Arizona. Which is basically (probably) why this Gohmert lawsuit was brought in this form that it was.