Sounds to me like this could have been an UTMA or UGMA account, but I guess that's water under the bridge at this point. But if it is an UTMA or UGMA account and your wife is no longer an minor she needs to take over the account. Her parents haven't done a tremendously good job of managing it if they have no idea of the basis of the securities in the accounts.
"My father in law thinks he showed the value of the account each year on
their taxes and paid any necessary income taxes each year as well."
He surely didn't "show" the value of the account each year in their taxes - nowhere is that required - and somebody had to pay the taxes in any case. If it was an UTMA or UGMA account that should have been your wife, benefiting to some extent by lower tax rates.
"He then thinks we should only owe on the difference between the value
when it was sold this year and when it was reported last year."
He's wrong. He didn't report the value of the account last year and that would only work in any case if the IRS taxed you on unrealized gains, effectively re-setting the basis every year. But you only are taxed on realized gains. So if a stock has been in the account for 10 years and has appreciated greatly over those years then when you sell the stock you pay tax on the gain between the selling price and the original basis from 10 years ago.
"The bank also did not report the type of gain or loss in the Form 1099B."
Since the concept of "covered" securities was introduced in 2010 that should mean that the securities sold were not purchased in the 2010 - 2016 time frame and therefore should be Box E sales.
"I am not sure how to file this, or if my father in law is supposed to be
filing it, or if anyone even needs to be filing it. Any help is greatly
appreciated."
Since a 1099-B was created somebody has to report those sales. The IRS will be looking for it. How is the account out of which the securities were sold actually titled? Whose Social Security Number is on the 1099-B?
After 20+ years of non-attention it's going to be pretty darn hard to come up with reasonable, supportable bases for the securities sold. Has the father kept 20+ years of broker/banker reports by any chance?
