No, you can't. To get a federally guaranteed loan, a law called FIRREA (the Federal Institutions Reform, Recovery and Enforcement Act) requires an appraisal from a professional appraiser. The Zestimate is an estimate of fair market value, a starting point for home buyers and sellers and anyone just plain interested in the value of houses. You can use it in negotiating, in judging market trends, and in calculating all sorts of things for your personal purposes.
For more information see Zillow and Zestimate.