How can I look up my iPhone specs (i.e.- how much storage it has on it) from my serial number?

From the serial number alone it cannot be done. While the serial number does identify a specific device, it does not have any info regarding the device's actual specifications.

If you have the something like the model number you can get the specs from it in websites like

everymac.com or apps like MacTracker.

That information is easily available in Settings > General > About.

Decode The Meaning Behind Your Apple Serial Number | BeetsBlog.