Short answer:
No, most browsers will not compress form data for client requests.
Long answer:
Yes, all browsers allow the client to send compressed form data. But since the browsers wouldn't compress the data for us, we've got to compress it ourselves.
Gzip uses the DEFLATE algorithm, which is publicly available and free to use. What can be done is to compress the form data on the client-side using JavaScript (download a JS Gzip library if you don't want to write one yourself), then send the compressed data to the server through either GET, POST, or PUT using XMLHttpRequest.
If you are in control of your web server, you could simply grab the data and uncompress it. If you are not in control, you will have to follow whatever policies set in place. For example, some web servers may require you to set a specific Content-Type, while others may not support it at all.
Lastly note that if your resource is a file that is already compressed, there may be no advantages in gzipping it. However if your resource is huge uncompressed data (e.g. a forum post of 80000 characters), the advantages are enormous.