Is the email all in an imap account?

I doubt it is imap. It is very probably pop. How could I get the person to check?

Help | troubleshooting lists the account names and types

They didn't have troubleshooting under Help (remember this is almost 10 years old) but I was able to determine it is a POP account.

I suggest installing tenfourbird on the old mac https://en.osdn.jp/projects/tenfourbird/

That is a Thunderbird 38 build for running on OSX 10.4 Once the update on the old mac has occurred it is only a matter of transferring the Stuff in ~Library to the new machine.,

The update should take but a few minutes.... note that passwords for accounts may not make the translation. (sissies with changes to security settings and storage over that 10 years.) So be sure you know them.

Thanks Matt. Can I install the current Thunderbird on the new iMac and move over the stuff in the ~Library folder? Or do I have to update the Thunderbird on the old iMac? Will it even update to the most recent Thunderbird on a Mac using 10.4?

update the old mac with tenfour. it is Thunderbird, but Thunderird it self dropped support for OSX 10.4 some time back, hence the community build.

Once it is installed it will leave you with a profile folder that is the same as the one used in Thunderbird V38 which is completely compatible with the one in the current version.

Alternatively you could install Thunderbird 1,5 on the new machine and go from there. Technically the mail data in ~Library will work with the latest Thunderbird version. So you might want to try that first. Sort of suck it and see if the result is acceptable. All the other stuff is really about making updates to profile data other than mail. That has changed not a wit since version 1