Learn Swedish: Kvinnan kommer att låta offerten skickas till elektrikern. - The woman will have the quote sent to the electrician.

Swedish forms this causative passive differently than English. Instead of saying 'have something done', Swedish literally says 'let something be done'. After the causative verb låta and its object (offerten), you must use the passive infinitive form of the action verb, which is the s-passive skickas (to be sent).

Could I use the 'bli' passive here and say låta offerten bli skickad?

While a native speaker would understand you, it sounds quite unnatural. The s-passive (skickas) is the standard and heavily preferred form to use in the låta causative construction.

The word offerten looks like 'the offer'. Is that exactly what it means?

It is a specific type of offer used in business and trades. En offert translates best to a 'quote', 'estimate', or 'bid' for a job or service. If you mean a casual offer (like a cup of coffee) or a job offer, you would instead use the word ett erbjudande.

Is it strictly necessary to include att in kommer att låta?