Can you pickup with a pokemon even if you don't send it out to battle?

Yes

You can also Pickup with more than one Pokemon, if you have more than one in your party. There's a 10% chance of finding an item (or around about that, based on a test sample of 200 battles and 19 items found with one Pokemon), so the odds of finding two items at once are very slim, but it is possible.

For the sake of straining your brain with a lot of numbers, the chance of finding items when you have more than one Pickup Pokemon in your party is calculated by cumulative odds.
For example, I have two and each one has a 10% chance of finding something after a battle.
There is a 1% chance that they'll both find something after the same battle.
There is an 18% chance that only one of the two Pokemon will find an item.
There is an 81% chance of finding absolutely nothing.

Mod Edit: Question Answered, so *Locked* (7/1)
-JsX™