How can I use an internal Zip drive?

Photos of drive

That drive uses an old (P)ATA / IDE controller. I have not seen this controllers/connectors on motherboards for years not. They have been replaced by SATA.
Maybe you can find an ATA controller card on which you can connect the drive. Another question is if you will be able to find drivers for modern OSes for that drive.

https://en.wikipedia.org/wiki/Parallel_ATA

Since it doesn't behave a like a hard disk, a USB converter will not help you. You need to install a full test machine (OLD though!) for this to work.

Iomega is now LenovoEMC and I was not able to find drivers for your drive. You probably will need a Windows XP computer hoping you have a CD with drivers. It is a device from year 2004...
Fro physical connection you can leave the pin set on master and connect it using IDE cable to IDE/ATA controller. 

If you have a Core 2 Duo PC around, check if it has IDE connectors. Many of them had both, SATA & IDE. This is also possible with older 2 or 4 Core AMD Athlon x64 CPU's.

Another option would be to use an External IDE USB Disk case.

Further you can get SATA to IDE Adapters, something like this:

https://www.newegg.com/riitop-35ide2sata-mndl-ide-to-sata/p/35G-0009-00054?Description=sata%20to%20ide%20adapter&cm_re=sata_to%20ide%20adapter-_-9SIA6V89YW2632-_-Product&quicklink=true

The problem could be that more modern OS's might not have drivers available to recognize ZIP drives, so you might also have to boot the PC with Linux.

Could I use an Iomega Zip 100 Portable USB Drive with a USB connection instead of the internal Zip drive?   Would it work with Windows 7 or 10?

You may try a Iomega REV USB drive, though I'm not sure if Windows 7 or 10 will recognize it without the drivers installed.

It probably wouldn't make much of a difference. Although I'd prefer the IDE version internally. It is faster & a more reliable connection. Drivers would be the same. So if ZIP Drives aren't supported by Windoze neither would work. The best chance would still be with Linux, as most Drivers are already built-in, & they don't usually get dropped by upgrades.

I'd go to the local computer recycler, say "I have ten dollars if you have a Windows 95 or 98 system", use it to pull the data off the drives and then recycle it again.

If you check a local computer recycling store (any nearby you?), you should be able to pick up a Windows XP computer VERY inexpensively.  That should allow you to connect the internal drive to the computer and get your data copied to a USB stick or external USB drive.