Learn French: Le chef demande si vous aviez cette compétence quand vous travailliez ici. - The boss asks if you had this skill when you were working here.

This sentence is reporting a yes/no question. The direct question the boss asked was probably something like, 'Did you have this skill?' Just like in English, when we report a yes/no question, we connect it with 'if' or 'whether', which is si in French. You drop the original question format (like est-ce que or inversion) completely.

Because this is reported speech, do we need to shift the tenses of the verbs that follow?

Not here! The introductory verb, demande (asks), is in the present tense. When the reporting verb is in the present, the verbs inside the reported question keep their original tenses. The boss was asking about a past state, so aviez and travailliez stay in the imperfect tense.

Why are aviez and travailliez in the imperfect tense rather than the passé composé?

The imperfect is used here because both verbs describe ongoing past states or background situations. Having a skill (vous aviez) is a continuous past state, and working somewhere (vous travailliez) sets the background scene. There is no sudden, completed action here that would require the passé composé.

Travailliez looks like a typo with that double 'i'. Is that correct?