Can you use entity possession (the trait on the WOL chases) on a bystander token that meets the trait's point value and combat symbol requirements (e.g., Skadi's Warbot)?

I'm going to say no since the Bystander Token doesn't have a dial to click too or any click numbers.

So since there isn't a dial to click, it isn't valid.

Yes, you can. It's perfectly legal. It meets all of the minimum criteria.

By your logic I can't assign an entity to a pre-WoS figure or have a pre-LoSH figure pick up a relic (the latter has been specifically ruled as doable).

Bystanders are single-click figures. They are, generally, understood to be on click #1. The exception to that might be the couple of double sided bystanders for Two Towers. I'm genuinely not sure how one would rule those if they were possessed.

Now that you mention it, I don't think you can assign an Entity to a pre-Web of Spider-Man figure. The standing ruling is that if a game effect specifies a "click number", which Possessed does, then a character without click numbers can't use or be affected by that game effect.

This originally came up with The Bat-Man's trait:

It was ruled at the time, and as far as I know it has never been reversed, that if The Bat-Man on click #3 attacked a pre-click number character on its first click, The Bat-Man would not get the modifier, because the target didn't have the little click number on its dial.

"Can't be affected by" is different from "cannot be assigned". You can assign the entity to those figures, it just does nothing.

This was the email I received. So no you can not.

Ugh, THAT illegal game state got investigated but the others didn't?

I e-mailed them Saturday asking them about this, and they finally got back to me this afternoon with the same response. Utterly ridiculous.