“Grudge,” from an old German word meaning “lament,” is a lot of fun to say. The noun “grudge” means “hostility or ill will against someone over a real or fancied grievance,” or the cause of that resentment, says Webster’s New World College Dictionary (Fourth Edition). “He bears a grudge because a woman was promoted instead of him” is one example.
The verb form means to envy someone because of something that person has: “She grudges him his higher salary.” There’s an adjective (“He gave her his grudging congratulations”) and an adverb (“She accepted his congratulations grudgingly”).
Add the prefix “be” to your grudge and things change. There is no noun “begrudge”; and the verb adds nuance: “She begrudges him his higher salary” sounds softer than “she grudges him his higher salary.” But when that resentment becomes an adjective, only “grudging” is allowed. “He gave her his begrudging congratulations” shows up frequently, though not in dictionaries. There’s an adverbial form, “begrudgingly,” but no adjective.
Enough people are using “begrudging” as an adjective, though, that it has reached Stage 4 on the five-stage Language-Change Index in Garner’s Modern American Usage, meaning there’s no need to “begrudge” anyone’s use of it.