Learn French: Pendant l'enquête, quelqu'un vole un document. - During the investigation, someone steals a document.

You always treat quelqu'un as a third-person singular subject, just like il, elle, or on. That is why we use vole here instead of a plural form.

Does quelqu'un change if the "someone" turns out to be a woman?

No, as a general indefinite pronoun meaning "someone", quelqu'un is always masculine singular. You do not need to change it to match the gender of the person.

Doesn't the verb voler mean "to fly"?

It does! The verb voler means both "to fly" and "to steal". You have to rely on the context of the sentence. Since a person cannot fly a document, we know it means "steals" here.

Could we use pendant que instead of pendant at the beginning of the sentence?

No, because l'enquête is a noun. You use pendant (during) before a noun, but you must use pendant que (while) when there is a full action with a subject and a verb following it.