How do I get my three year old child to tell me what she wants?

When my daughter wants something, sometimes she'll ask for it, but usually she makes us play 20 questions to figure it out. She can say any of things she needs, but she'll just stare at us expectantly, and get ever more frustrated if we can't read her thoughts. Sometimes she'll mime things rather than speak. If you ignore it, she has a meltdown. Her preschool teachers described her as "willful." We always encourage her to speak, but she doesn't cooperate. She has never parroted, but uses words when she's ready. She is slightly delayed in her vocabulary but uses short sentences at her discretion. For example:

"I gotta go to school!" "I gotta eat!" "Goodnight mama." "What's that?" "I don't want it!"

We're getting her tested in May for developmental delays to qualify her for additional schooling. We were wanting to see if anybody has had a similar experience, and how it turned out. It's really stressful, and we're getting tired of this game.