Can I open Illustrator files in Corel Vector?

No, but if the Illustrator files are saved with the option “PDF compatible file” checked you just need to rename your .ai file to .pdf. If this is not the case, you can export an SVG or a PDF from Illustrator and import it back into Corel Vector, or if you don’t have Illustrator available, you can also use an online tool like https://cloudconvert.com/ai-to-svg.