Can Vuejs be used on sharepoint 2013 on premise or only with the SPFx?
If so, would appreciate some guidance.
SharePoint 2013 doesn't support SPFx yet.SPFx works for SharePoint Online and also for on-premises (SharePoint 2016 Feature Pack 2 and SharePoint 2019).
While you could use vue js framework in SharePoint 2013(add-in or Content Editor web part), Check the sample project.
Yes, you can use the VueJS with SharePoint 2013 on prem without SPFx framework. As Lee mentioned it is not supported with SharePoint 2013.
You can create Modern Webpart using it, the below blog explain in the detail with the demo.
Another gitHub example "Vuejs-Highcharts-SharePoint" and also check this quesiton from stack-exchange which resolve the issue to getting list data Stuck on using vuejs to create a table in SharePoint 2013