Learn Italian: Si comportano come bambini, fanno costantemente dispetti. - They behave like children, they constantly do spiteful things.

You would say Ce l'ho con loro perché... The pronominal verb avercela (to have it out for / to be angry with) is always followed by the preposition con when indicating the person you are mad at.

Is fare dispetti the only way to translate "doing spiteful things"?

Un dispetto is a specific type of action done deliberately to annoy, tease, or inconvenience someone. It is exactly the word Italians use for the petty, annoying things children do to each other (like snatching toys or pulling hair), making fare dispetti the most natural fit for this context.

Why is there no article before bambini? Could I say "come i bambini"?

You leave out the article here because come bambini acts almost like an adjective describing the quality of their behavior—they are acting "childishly." While come i bambini isn't strictly wrong, omitting the article is the standard, idiomatic way to make this comparison.

Why is the adverb costantemente placed between the verb and the noun?

In Italian, adverbs of frequency and manner (like costantemente, sempre, or spesso) very commonly sit right between the conjugated verb (fanno) and the direct object (dispetti). You could put it at the end, but right after the verb is the most natural word order.