Because your document has default namespace declared in the most outer element you have to repeat that namespace on every child element to avoid adding additional empty one.
Change request
and param
elements declaration to contain "http://www.ebay.com/marketplace/search/v1/services"
namespace
XmlElement request = (XmlElement)body.AppendChild(doc.CreateElement("findItemsByKeywordsRequest", "http://www.ebay.com/marketplace/search/v1/services"));
XmlElement param = (XmlElement)request.AppendChild(doc.CreateElement("keywords", "http://www.ebay.com/marketplace/search/v1/services"));
With these changes your code produces following XML:
<soap:Envelope xmlns="http://www.ebay.com/marketplace/search/v1/services" xmlns:soap="http://www.w3.org/2003/05/soap-envelope">
<soap:Header />
<soap:Body>
<findItemsByKeywordsRequest>
<keywords>harry potter phoenix</keywords>
</findItemsByKeywordsRequest>
</soap:Body>
</soap:Envelope>