Thanks everyone for the suggestions.
Logging into the full Facebook site and then going to m.facebook.com gets me into the latter, but I still can't type anything (i.e. to make a post).
touch.facebook.com seems to work though. Thanks! That interface is even nicer than m.facebook.com is. Ironic, since it seems to be meant for touchscreen devices, which the Kindle isn't.
I figured it was some sort of browser bug. I tried writing a few little HTML pages with text input, and I haven't been able to come up with one that doesn't work on the Kindle, but apparently Facebook has.
Thanks again, Stefan
PS. I reported another bug to Amazon, which is that the browser refuses to follow links that have a "target" attribute. The browser complains that it doesn't support multiple windows. I don't expect them to honor the "target" tag, but I do expect them to follow the link.