Which of the following is grammatical?
- I am deputizing for him/her.
- I am deputizing him/her.
I think in the case of covering you have to use:
I am covering for him/her.
...or do you?
Merriam-Webster shows a usage of deputize that somewhat agrees with your first sentence: I deputize for the newspaper's editor on the weekends. But it sounds strange to me. I'd say that I was acting for or covering for someone not available to do the work at the moment, or that I had been authorized to act as that person's agent. I associate deputy with legal positions, like deputy sheriff and deputy district attorney. There may be a distinction in British and American English, though.
I am deputizing him/her means that you are authorizing someone else to act as a deputy.