The list of fonts in a word file is stored in an XML file called fonttable.xml. To get this file, you need to convert the word file into a package (zip file) and parse the contents of fonttable.xml.
Note: You need to add 2 references: DocumentFormat.OpenXml.dll and WindowsBase.dll to your project for handling word processing.
Following code handles getting the file from document library using CSOM and reading the file to get the fonts and stores them in a generic list.
ClientContext ctx = new ClientContext("http://siteUrl");
Microsoft.SharePoint.Client.ListItem listItem = ctx.Web.Lists.GetByTitle("Documents").GetItemById(itemID);
ctx.Load(listItem);
ctx.ExecuteQuery();
Microsoft.SharePoint.Client.File file = listItem.File;
ClientResult<Stream> data = file.OpenBinaryStream();
ctx.Load(ctx.Web, w => w.Url);
ctx.Load(file, t => t.ServerRelativeUrl);
ctx.ExecuteQuery();
WordprocessingDocument wordprocessingDocument = WordprocessingDocument.Open(data.Value, false);
var fontTable = wordprocessingDocument.MainDocumentPart.OpenXmlPackage.Package.GetPart(new Uri(@"/word/fontTable.xml", UriKind.Relative));
XmlDataDocument xmldoc = new XmlDataDocument();
XmlNodeList xmlnode;
int i = 0;
Stream fs = fontTable.GetStream();
xmldoc.Load(fs);
xmlnode = xmldoc.GetElementsByTagName("w:fonts");
var childNodes = xmlnode[0].ChildNodes;
List<string> allFonts = new List<string>();
for (i = 0; i <= childNodes.Count - 1; i++)
{
allFonts.Add(childNodes[i].Attributes[0].Value);
}
ctx.Dispose();