。
She always wants to put on airs.
Literal
She [topic-は] vanity [object-を] shows-signs-of-wanting-to-stretch.
見栄を張る ('to stretch one's vanity') is the standard idiom for 'putting on airs, keeping up appearances, showing off so people will think well of you.' It often carries a tinge of mild social criticism — the vanity is visible to everyone but the person performing it. The desiderative ~たい becomes ~たがる when you're describing someone else's wanting: Japanese reserves bare ~たい for first-person desires (or close empathic positions) and switches to ~たがる ('shows signs of wanting to') for third parties — a small built-in respect for the limits of mind-reading.