I think it's probably just not worth the risk. Anything you catch is going to have the mark of it's habitat on it - the same reason that you wouldn't necessarily feed your axie worms from a garden that you didn't know what the soil was like - aside from the venom (which WOULD kill your axolotl) often common spiders have a poisonous substance on their skin to deter predators. Also I don't know too much about spiders (except for when I see one, I have to call the fella!) but that red and black one in the picture automatically spells danger for me, and most likely your axie!! Not to mention all the parasites, pathogens, pesticides from the environment you found it in...
I know that as humans, it's hard to look at those awful earthworms and imagine eating those for the rest of your life. But when you see your axie eat one, does he look up at you and go "Really? Another worm? Again? Ah well, ok, I'll force it down" whilst he watches a spider across the room and prepares to pounce, or watches you eat a pizza and think "Oooh, I'd like a bit of that!"
... or does he do as mine, and everybody's who has ever kept one for a while does and swim straight up not even thinking about what he might bump in to and snap at it - making it disappear right away! And even look around for more when you're feeding his friends?!
Think about what they would eat in the wild. They don't have the means or the whereabouts to chase spiders. That's for a very good reason - that's not what they really should be eating. I'm not too sure about the logistics about how they catch worms in the wild - though I'm sure someone more experienced could tell you- but I do know that worms have been used throughout time to catch fish which also love them.
They are highly packed with nutrients, and you can vary the diet that the worms themselves eat for your axie (If I feed the worms exclusively on carrot peelings, for example, my little worm-muncher seems to be a bolder colour and very energetic, but if I feed the worms bits of lettuce then Puck seems to be going to the toilet more often...!) but anyway, also, more to the point, axolotls LOVE them. Why would you waste time chasing spiders and trying to get your axie into them, which could be fatal... When you have all the nutrients you need in your little ice-cream tub, or bait shop, or even pesticide-free back garden full of gloriously juicy, fat, actually quite disgusting worms!
As ever, if you must vary the diet, try a sinking pellet (which I think are messy and can foul the water) or some live river shrimp (which you CAN catch yourself, if you quarantine them for a few days!) or minnows... And a little bit of beef heart, if you cut it into matchstick strips and make sure it's rinsed, goes down great as a bit of a treat. But worms are where it's at.
Please please please stay away from the spiders. It's only going to lead to trouble!
