Disclaimer: I am not a lawyer.
I don't know.
If a company is dissolved before lawsuits or charges are filed, the argument could be made that the entity in question no longer exists and the filings are invalid. Just like you can't sue somebody who's dead. It might not hold up in court but I wouldn't put it past some very expensive lawyers to try it anyway because it might work.
This article says that "it depends." There might be a period of time after a company dissolves that it can still be sued, namely, if the legal process to go about it wasn't followed precisely. If there are no assets remaining sometimes the former owners can be sued. There is also the question of whether or not you'll spend more on a lawsuit than you'll get from the settlement.
I just realized something: Most of the time when talking about stuff like this, people seem to implicitly be talking about getting some money out of it (as punishment, maybe). Rarely do folks ever talk about suing for the express purpose of preventing the thing (in this case, selling customers' genomic information to third parties) from ever happening.
This article talks about suing for undistributed assets. Suing to get your genomic data back and verifying that it's been destroyed before it could be sold to anyone else is a possibility. It also talks about suing shareholders; if 23andMe is being delisted that seems like a legal gray area to be exploited: If a company is delisted are there still shareholders? Logically, yes (people hold worthless shares of stock in a company that doesn't exist anymore) but legally? It might be state-dependent as this article suggests (per Favila v. Katten Muchin Rosenman LLP (2010) 188 Cal.App.4th 189, 213).
Maybe under a quiet title action to get the genomic data back?
But hardly practical.