I think that’s a lovely idea - and congrats!
BTW, I moved your post to Everything Else because Made on a Glowforge is where to post finished projects
Clock movements can be purchased locally at places like Michael’s, online from Amazon, or specialists like Clockworks. They all require batteries, or wired power, or solar, or gravity - but power in some way shape or form. Batteries are easy and last a long time.
If you want to do it as a 3D engrave I would suggest doing it on the backside of piece of transparent/translucent acrylic so the front of the clock is smooth. In that case you don’t need a vector, that requires raster art - but it’s easy to turn one into the other in Inkscape (or any design program).
Doing it as a layered cut would also look gorgeous - I’d suggest the doing the clockface out of a light wood like maple, and the heart out of a medium one like cherry - you could do the numbers out of something dark like mahogany, and get either brightly coloured or black clock hands.
I look forward to seeing what you come up with!