I want to cut down a crab apple tree and let it sprout from the trunk.

I would late fall when the saps not moving so fast.

According to this article, the way to propagate a crab apple tree is to remove a sucker and plant that. The other issue is knowing whether your tree was grafted onto a different base. This article might help but it really doesn't say how to figure out if your tree was grafted. Do you care if the new tree is different from the one you have? Even if it's a sprout on the tree, if it was grafted you might get a different looking tree. If you want to keep the one you have, I think it would be best to call a professional - at least for advice.

http://homeguides.sfgate.com/propagate-trees-root-suckers-41343.html

Good luck with your crab apple tree.

And why do you want to kill a perfectly good tree just to regrow it from a side sprout/sucker? Leave the tree alone and plant a sucker.

Https://ask.extension.org can advise you with info for your grow zone

Late fall would be best :)