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If png is acceptable for you, there might be a way to do it. Keep in mind that I haven’t tested this, but it should work by the looks of it.
Looking at the documentation of vue-chartjs, here it says that you can access the chartjs object with:
this.$data._chart
From there, you should be able to use its method for saving the chart as a png – toBase64Image():
this.$data._chart.toBase64Image()
Source:stackexchange.com