Learn French: Si vous étiez ambitieux, vous auriez accepté ce rôle. - If you were ambitious, you would have accepted this role.

This is a mixed conditional. It links a present state to a past result. Because being ambitious is an ongoing personality trait (present condition), we use the imperfect étiez after si. But the opportunity to accept the role has already passed, so we use the past conditional auriez accepté for the result.

If I am using the formal 'vous' to speak to a woman, how does the word 'ambitieux' change?

It changes to ambitieuse. Even though vous is the pronoun, adjectives must still agree in gender and number with the actual person you are addressing. If you were speaking to a group of men, it would stay ambitieux since adjectives ending in -x do not change in the masculine plural.

Are there any mandatory liaisons in this sentence?

Yes, there are two! You must pronounce a 'z' sound between the subject pronoun and the vowel-starting verbs that follow it: vous (z) étiez and vous (z) auriez.

Does 'rôle' only refer to acting, or can it mean a job position like in English?

It translates perfectly to the English business sense of the word. While it certainly means an actor's part, ce rôle is very commonly used in professional contexts to mean a job, position, or specific set of responsibilities.