Would I be correct in assuming that if I simply bought a new 25 pin SCSI card, that would simplify the entire process?

Robricc and muzza are right: the process is already simple.

You've got no internal devices on the SCSI bus, right? Just the card, the tapestreamer, and the scanner, right? In that case it's as robricc drew it:

SCSI card (terminated; modern cards self-terminate automatically)
|
| SCSI cable
|
Tapestreamer (not terminated)
|
| SCSI cable
|
Scanner (terminated)

If the scanner doesn't have a DIP switch to enable termination, you'll need a separate physical terminator on the scanner's other SCSI connector (it has two, right?)

The two SCSI cables in the picture can be either 50-to-25 or 50-to-50 as necessary. 25-pin SCSI is just a different pinout for exactly the same thing as 50-pin SCSI. You also have to make sure that the card, tapestreamer, and scanner are all set to different SCSI IDs (the card is traditionally 7 but you can choose any two of 0-6 for the other things; they don't have to be daisychained in ID order).

Peter