Format
I'd give your client an RGB JPEG of your logo saved at 300dpi.
Before anyone shouts, I know that screen based images are 72dpi, but this way the website designer has some flexibility with scaling it without losing detail and can then re-save the image down to 72dpi once the size has been agreed.
Unless of course, you know exactly the size that it is needed on the site in which case you can skip the middle step and save it at 100% at 72dpi.
You may be able to keep the size of the logo very small by saving it as a GIF file, but this depends on the complexity of your logo.
Hope this helps.