Can I get a lightning with chanelling trident during a storm in a no-rain biome?
Yes it is possible.
I loaded up a creative world, enchanted a trident, and found a desert. I changed the weather to thunder and threw the trident at a sheep. The lightning was summoned. I tried it in clear weather and there was no lightning.
Also, because I was in the desert there was no rain and the trident worked anyway.