Third party software patching as a non-admin

Yes, it's possible, but it depends on the application(s) that need updates and how that needs happen. Most applications can be handled via the Patch Management feature or by creating/scoping policies to handle updates after you have created/uploaded the package. Others could be managed by configuring updates in profiles or setting permissions on the application.

1st thing that comes to my mind is disable the auto updates in the software (probably through deploying a plist or preference file) , then package the updates and make them available in jamf self service which would run as your  jamf management account. I just dont think patch management would give you the scheduling you are looking for. 

A little more information. This software is not in the App store. It is downloaded from a website and updates are triggered at random times. 

There will be some effort on your part to get things going, but you could deploy the applications with their in-built update feature disabled. Then, as updates are published by the vendor, package and distribute the software via Jamf either as a standard policy or as a Patch Management patch policy (if the app is supported). The advantage of the patch policy, is the definition knows how to handle if the app is running on the client, can easily ignore clients that are already patched (or ahead) of the version you are deploying, and can notify users updates are available. A standard policy is great if you want the "patch day" approach where central IT pushes updates on a set cadence and forcibly installs updates with fewer native controls for user interaction and you have handled the case where the software is currently running and how to quit/update the app yourself.