My company has a support contract from Microsoft so we are able to download XP patches - but I don't know how to push them to our machines. Can I just create a custom patch? If it's possible to deploy them this way, I'd like some guidance on creating custom patches.
I am guessing you're actually using the Shavlik Protect product, not Shavlik Patch for SCCM.
Any XP patches that were available prior to EOL and are still downloadable from vendor can be deployed in Protect like normal. Anything after the EOL, we do offer extended support (see this document) or you can create custom patches yourself to deploy.
This document provides an example of how to create a custom patch: