How can I scan from my Canon Pixma MP140 to Mojave? I tried using Image Capture, but there are no devices found.

I tried using Preview, but the "Import from Scanner" menu item is greyed out. I also tried this answer with no luck.

Then I tried using MP Navigator but it's not compatible with the Canon MP140.

Finally, I  tried using TWAIN-SANE but I get an error: "scanimage: no SANE devices found.”

Please don’t tell me that the only option is to spend $45 on VueScan software to get my scanner working!  How can I get this scanner working on Mojave?
This question is a bit scattered, but if I understand it correctly, you can’t get this scanner to work with your Mac.
This scanner is EoL - End of Life
Looking at the support page for this particular multi-function device, the last version of macOS that was supported was OS X 10.5.x (Leopard) and that version was released in 2009 and support ended in June 2011. The drivers were released for both Intel and PowerPC architectures, the latter being EOL itself since 2006; just about a decade and a half ago at the time of writing this answer.
Development has (obviously) stopped on this product.  If it were to have TWAIN drivers, it probably would continue to function.  Unfortunately, it requires it’s own drivers provided for by Canon.
As far as the Intel drivers go, being that they are so old, they are 32-bit drivers.  If you’re attempting to use this on Catalina, this will be a non-starter as Catalina doesn’t support 32-bit apps or drivers..

Please tell me the only option is not really to spend $45 on VueScan?

It’s possible.  I can’t speak to the viability of the software.  While I’m a huge proponent of extracting every last ounce of usability out of a computing device, there are certain types of products that become much more expensive to maintain than it is to simply purchase new.  Scanners and laser printers have consumable parts like pinch rollers (usually made of rubber), stepper motors, and sensors that do fail.  When these items fail especially the rollers - they become dry and hard and no longer “grips” the paper, I usually dispose of the device unless I can find a quality “rebuild kit.”
Perhaps you can get it working again, but keep in mind, you’ll be spending $45 and in the end, you’ll still have an obsolete scanner that’s nearly a decade and a half old.
I’d spring for a new scanner.