If you plan to sue, you have to identify a legally cognizable injury in the complaint. So I guess you are asking us to first devise a plausible injury to the candidate coming from the IRS action, and then see if that really gives them standing. At the outset, it is also important to fix on a reason why tax exempt status was not revoked (there is a separate law, the Church Audit Procedures Act about enforcing the Johnson provisions,
see Branch Ministries v. Rossotti), i.e. was there a complaint, was it investigated... did the IRS say "Yep, a violation, but we don't care"?
The case Allen v. Wright, 468 U.S. 737 is relevant to this attempt. Racially-discriminatory private schools are not eligible for tax exemptions granted by the IRS: plaintiffs alleged that IRS procedures for vetting applicants on this ground were insufficient. The Supreme Court held that plaintiffs did not have standing to bring this lawsuit. The harms alleged in the suit are summarized as follows
First, they say that they are harmed directly by the mere fact of
Government financial aid to discriminatory private schools. Second,
they say that the federal tax exemptions to racially discriminatory
private schools in their communities impair their ability to have
their public schools desegregated.
These harms are analogous to ones that might be raised in your scenario. The alleged harms in the case were not well-enough explained that the court had to do a lot of conjecturing to decide whether there was cognizable harm. It was observed that
This Court has repeatedly held that an asserted right to have the
Government act in accordance with law is not sufficient, standing
alone, to confer jurisdiction on a federal court
with various citations ("Ex parte Levitt, 302 U.S. 633 (1937). Respondents here have no standing to complain simply that their Government is violating the law"). You can't just sue the government because it isn't enforcing the law.
The problem is that the candidate is not obviously harmed by the Government giving tax relief to a third party. A suggestion is made in the opinion as to how such an argument might be sustainable:
The diminished ability of respondents' children to receive a
desegregated education would be fairly traceable to unlawful IRS
grants of tax exemptions only if there were enough racially
discriminatory private schools receiving tax exemptions in
respondents' communities for withdrawal of those exemptions to make an
appreciable difference in public school integration.
Plaintiffs did not make any such argument.
We would need to identify a right of the plaintiff-candidate which was trod upon by the government, in not withdrawing tax-exempt status of a third party, one who exercised their First Amendment right to express support for or against a political candidate. Whereas in the education case there is a clear right (the right to receive an education irrespective of race), there is no "right to be elected", there is at best a "right to run for office" and not a "right to win". Taxing the ministry is not likely to enable a candidate's "right to win".