You need a software indeed to make a logo, but the client doesn't need to have the software as well. You design the logo, save it as an image, upload the image to a website that is always online (for example your own site) and insert the image into the signature (from
http://www.yoursite.com/the_image.jpg). You do that like this : Tools -> Options -> Mail Format -> Signatures. You instruct the client to do the same steps and that's it.