Is it obligatory to give out a third of the salary for the sake of Allah—as the questioner previously vowed—after Allah has blessed him with a job, or is it permissible for him to give out whatever amount of money is available to him?

If your intention behind what you said was a vow (nadhr), then you must fulfill it and give one-third of your salary as charity for the sake of Allah the Almighty, based on the hadith: "Whoever vows to obey Allah, let him obey Him." If you did not intend a vow, then nothing is binding upon you, but giving in charity what you intended is better for you.