I like using the vector feature to draw on Scratch and it's really helpful. However, I can't save parts of a costume or a sprite to my computer to animate on a separate app.

You can already save costumes/sprites to your device by right-clicking on the costume/sprite and clicking “Export”. However, sprites will be exported as a .sprite3 file, so you might want to export the costumes instead, which are exported as .svg files.

Thanks for your help! I'll go use that.