Will Verizon work with a Xperia phone?

Yes, as of February 15 2011 Verizon began to sell the Sony Ericsson PlayStation phone. Though I would look at your local Verizon wireless store depending on the location they might not sell it. So it is always better to go many times to the local retailer to ask before buying for when a new product comes out they have to settle how to best sell it and will inform each store in time with updates on the product.