He's not particularly ticklish, no - not that it would make much difference if he were, since he came on this little adventure in full armor, and it's difficult to feel much through the leather.

What matters slightly more in this scenario is Vanyel's past history with spiders. Cave spiders in Skyrim tend to only come in one flavor: frostbite spiders, which are huge, poisonous, and extremely aggressive. So his automatic, instinctive response to the swarm of spiders is to fling a rather large fireball between the group and the heart of the swarm.

Some few might get burned, but he's hoping the fire will drive most of them off without the need for a fight - and, more importantly, without any of the party getting bitten.