Are the verbs transitive or intransitive Elaine won election and began fulfilling her campaign promises.?

* "won" is transitive. It takes a direct object: "Elaine won election."

* "began" is transitive. It takes a direct object: "Elaine began fulfilling."

* Note that "fulfilling" itself is a gerund, acting as the direct object.

In short: Both verbs in this sentence are transitive.