I'm trying to open old 2002-2006 .tax files but no longer have the software on my computer. Is there anyway to open these files?

If you did not save those old returns as pdf's then, no, you will not be able to open them.   There is a service --Taxprinter.com -- that is not affiliated with TurboTax which you might want to explore.  We do not know much about it or how much their fees are.

I still have the software on my PC for a few years of my tax returns.  When I try to log on, I get the familiar Update required and when I do an update, the tax return disappears.   I thought it was just last year's return but it's all of them.  Gone.  There is no way to open a return without accepting the update.

I wanted to see what would happen if I tried to reinstall 2019.  It wouldn't even accept my license code, which I still have.

Sorry---TurboTax no longer supports the software for any year earlier than 2021.  TT will not provide updates or the state software for years earlier than 2021.  You will not be able to install 2019, even with the license code.