Where can I buy a replacement Corning optical patch cable for my wall to modem connection?

Check the below reply.  He ordered one from Amazon.  You just need to make sure you get the correct ends and mode.

https://forums.att.com/conversations/att-fiber-installation/inside-fiber-optic-cable-length-from-fiber-jack-to-att-modem-limitations/5ff8d6d7deda6213d21a1265?commentId=601d9b7664100a03adc829bb

The cable in the link is not the same as mine.

The cable might look different but it is the ends of the cable that is important...

You have a 320 gateway?

With fiber connection from wall plate to SFP module of the gateway?

Need single mode fiber cable SC/APC to SC/APC

IF you are connected to the RED port of the gateway, this is not a fiber cable but an ethernet cable from the ONT which may be in the room or from a different location inside the residence. 

Pics always help when requesting specific information...

OP said "corning optical" though he did also say "cable".  I decided the two fiber-y words outweighed the one coppery word. :)

I figured it must be single mode, but I didn't know how to spec the ends.

Here's the end

This is the label from the cable bag the AT&T installer gave me.

..

Yes green ends are SC/APC... other types of ends would be a different color.

https://community.fs.com/blog/fiberstore-cables-how-to-identify-the-color-code-of-fiber-optic-cable.html

Connector Color Code

Connectors are also a part of the fiber color code. Since there are different polish styles of fiber end-face, the connectors of fiber jumpers and the mating adapters are color coded for identification. However, the advent of metallic connectors like the FC and ST made connector color coding difficult, so colored strain relief boots are also used. The boot color may vary among manufacturers.

How do you remove these Fiber cables from the router and wall jack?

Verrry carefully and only if absolutely necessary.  It's not difficult to damage the fiber and you'll be down for however many days it takes for AT&T to roll a truck and to create an invoice for $99 or $149.

These connectors just pull out (don't pull on the cable or strain relief itself!).  If you really and truly want to risk it, look at some YouTube videos that show this sort of cable so you get the idea.