What actually breaks
A draw is a circle, or several. Each person buys for the next one along. When one person leaves, two things happen. The person they were buying for now has no Santa. And the person who was buying for them now has nobody to buy for.
Notice that the fix is already there. Join those two ends. The Santa who lost their giftee takes over the giftee who lost their Santa. Everyone else's match is exactly what it was, and nobody has to be told anything except those two people.
If it happens before anyone has bought
Remove the person, join the ends, and tell the one Santa whose giftee changed. That is the whole procedure. Do not announce it to the group. The person who left does not need a message either, unless they ask.
If the person who left was the only one in their department under a within-department rule, the join might break that rule. Drop the rule for that one pair rather than redrawing forty people.
If it happens after people have bought
The Santa who lost their giftee has a gift with the wrong person's name on it. Three options, in order of preference. If the gift is general enough, it goes to the new giftee and nobody minds. If it is specific, the Santa swaps it or buys again, and the organiser quietly covers the difference if that is a hardship. If the person left in the final two days, the organiser buys the one missing gift themself and says so at the opening. That last one happens more than you would think and is always fine.
When a full redraw is right
Almost never. A full redraw throws away every gift already bought and every wishlist conversation already had. It is right in exactly two cases. The draw was wrong from the start, because a rule was missing or someone was left out. Or more than a quarter of the group has gone, usually because a whole team was pulled onto something else.
If you must redraw, do it once, do it early, and tell people to hold their shopping until the new match is out. If you have a tool that can keep the unchanged matches and their conversations through a redraw, use that instead, because most people's match will not change and they should not lose their chat.
The person who is in but has gone quiet
Different problem. They have not dropped out, they have just not opened the link. Send one reminder to the group, then one direct message to them with the link. If they have not opened it four days before the exchange, treat it as a drop-out and remove them. A silent person on the day is worse than a removed person a week before, because their giftee turns up and gets nothing.