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$refs.vta
only returns the v-textarea element, which is a wrapper element containing its own refs which includes the inner <textarea>
input element. You can access the inner element to get selectionStart and selectionEnd like so:
this.start = this.$refs.vta.$refs.input.selectionStart;
this.end = this.$refs.vta.$refs.input.selectionEnd;
Source:stackexchange.com