Learn French: Elle demande si vous avez les qualifications nécessaires pour ce poste. - She asks if you have the necessary qualifications for this position.

Because this is an indirect question. When reporting a yes/no question, French uses si (if or whether) followed by the normal subject-verb word order. You do not use inversion or est-ce que after si in reported speech.

Would si drop its last letter if the next word started with a vowel, like elle?

No, the word si only drops its -i when followed directly by il or ils (becoming s'il or s'ils). Before any other word, even those starting with vowels like elle or on, it remains si.

Why doesn't the tense of avez change to the imperfect here?

Because the introductory verb demande is in the present tense. When reporting speech introduced by a present tense verb, the tense of the original quote does not change. If her original question was in the present tense, it stays in the present tense in the reported version.

Does demande mean she is demanding or insisting on the qualifications?