Snow and Road Closures in mid to late April?

If you had any idea how much traffic goes to/from the Alps in the dead of winter, you'd be shocked. These are major tourist destinations (ski destinations) so keeping the roads open is critical And late April is hardly the dead of winter, its springtime.

If you intend sightseeing along the way you might find that some minor roads across some of the more remote passes are closed, but there will always be a more major road open.

If you are driving from Avignon to Chamonix in one day, you will probably want to stick to the faster roads via Valence, Grenoble and Albertville, where you will not encounter any significant mountain passes.
A longer option, much more scenic, going via Nyons, Serres and the col de la Croix Haute to Grenoble (then same as above) is passable year-round, but takes about 6.5 hours without stops.

If you have more time than one day, you could be tempted by a more easterly route via Briançon, but passes like Galibier, Iseran, Madeleine, Roselend will not be open, so I would not bother going that way.

To further Simon's point, we were driving from Lyon to Chamonix in December 2021 and there was a big snow forecasted for the days before we were making the drive. I was so nervous - there were four of us and all our luggage absolutely packed in a little economy car with a manual transmission! I should not have worried. Road was completely clear, rest stops were open and accessible, and the drive was absolutely beautiful. We arrived in Chamonix, dropped our bags and immediately turned in the car so we didn't have to worry with parking. We were so lucky with that freak early snow - Chamonix looked like a winter wonderland.

Thanks to all. Very helpful. Balso—appreciate the details about alternate routes.