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.
SharePoint 2013 Modern WebPart using VueJS
Another gitHub example "Vuejs-Highcharts-SharePoint" and also check this quesiton ...
Can't write a comment, so here goes the "answer":
SharePoint enforces strict mode as far as I know. Perhaps check if your resulting script does meet this requirement?
You say you use polyfills, but maybe you used the wrong target when transpiling typescript? IE doesn't support ES6.
Other things I could imagine would be a race condition, that causes a ...