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.
The layering options are endless! You could do a light color for the ribs on the outermost part, another color wood for the middle of the ribs, and a third even for the breast plate. Glue all of that up on a lighter colored wood backing or make it pop with a dark wood backing. So many options!
As far as clock parts, hobby stores are great but you can also check hardware stores, woodworking stores like Rockler Woodworking, or as easy as Amazon. As far an option without batteries Amazon might be your best bet. I’ve only seen the standard battery option at the listed physical stores.
Be sure to share what you decide to go with, I look forward to seeing it! Good luck and be sure to have fun with the process! The people here in the forums are some of the most helpful people I’ve (virtually) met.