Considering that so many people are opposed to using deodorants for themselves, due to concerns about the chemicals, I strongly would discourage the use of a recycled/repurposed deodorant tube. However...
I do have a creative alternative: Get a baby bottle, complete with ring/cap. Get a small ball to fit just inside the neck/rim of the bottle, and put the ring over it to hold it in place. Since it will not form a tight seal, there will be enough leakage for the dog to lick and get some yum yums out of the bottle.
Or you can recycle one of those frozen yogurt push ups-- the cardboard tube with the plastic disc and stick you push up and lick... This would work well for frozen broth and peanut butter, spreadable cheese, etc.
Be yourself. Then no one can ever tell you you're doing it wrong.