... I would use a basis of $0 (which any tax software should be able to handle). I don't think that it will be worth the hours of your time and the hassle to establish a more accurate basis. ...
Agreed. I did find a price going back to 1973, of $1.67, Aug 2024 is ~ $80. I'd assume the 1968 price is lower. So the delta of using something like $1 cost basis and $0 cost basis means you might have a gain of ~ 1/80th more with the 0$ basis - and that might be taxed at only 15%, and some/all may be zero tax rate anyhow, depending on your other income. Not worth the foort most likely.
Dividends are a larger portion than that, so if you have a record of those, it would be reasonable to add those to the cost basis (reducing the total gain). Even just add in the divs you have a record for, even if it isn't all of them. I don't see how the IRS can question that (well, they might
question it, it's unlikely, but you'd be on solid ground). Yahoo has divs going back to 1973, but you'd also need to know how many shares you had, if you don't have the div records. Even the last 10 years would be roughly $20/share in divs (~ $2/share annually) - so now you are talking something that might be significant.
So first, with zero basis, how much tax is due (add and remove it in your tax program, to see the delta with all other taxes/deductions entered)? If it is insignificant, just use $0 basis. If it is significant, see what you can find for divs.