Let's say a Supreme Court Justice resigned today because of health reasons or whatever...

[Polybius]

What happens to the cases coming up in April, May, or June? Will their vote be on record, or will any case that's decided after they leave be with only 8 Justices?

[Walleye]

1. I think they will decide with eight justices, the way they did when McConnell stole Garland's seat

[Buckeyeblue]

2. I don't believe so

I think you have to be part of the court when the order is issued.

He won't resign. Ever.

[bucolic_frolic]

3. Don't get your hopes up. The only reason he'll resign is if his owners decide he'll talk too much

He was only allowed to sit and be silent. Obviously.

[unblock]

4. Posthumous judgments/opinions have happened in federal courts of appeal

But not, afaik, at the Supreme Court.

I don't know if this is policy of if the question has ever come up.

[Polybius]

5. The right-wing needs him until at least June

That's when the abortion and NY gun-control cases will be decided. I expect a 5-4 decision on the former, and a 6-3 one on the latter.

[Beastly Boy]

7. A purely hypothetical question.

And a purely hypothetical answer would be: eight justices will make the decision. Not sure what happens in the event of a tie. I believe the case goes back to the previous court or the decision of that court prevails.

But I very much doubt that Kavanaugh will resign any time soon.

[Lettuce Be]

8. They continue with the justices they have

You can't vote on a case before hearing it, so he couldn't vote. They had only eight the entire time they left a seat vacant because it was "an election year." Still hard to believe they were able to get away with that.

[DetroitLegalBeagle]

9. If they already voted, their vote stands

If they haven't voted, then the court votes with whatever members are left.

[Princess Turandot]

10. They vote at the Friday conference immediately following the oral arguments for a case..

Then the decisions gets committed in writing by whomever is assigned to writing the opinion, in a process that can take awhile.

So, if it's after oral arguments/conference for a case that the hypothetical justice departs, their vote would normally count. If they departed before arguments on a case, they are no longer involved in the outcome.