Are there any ongoing projects trying to develop a widely accessible WebCL-powered mining software? I've heard of two in the past, but their websites don't seem to be working any more.
While not directly a WebCL miner there is a miner which uses OpenCL via Java Webstart so is accessible to any internet user with Java installed and apropriate hardware.
The site is both a pool and a custom java miner. The pool can be used with other miners however the BitMinter miner works only with the BitMinter pool.
So let me summarize all the other answers:
BitMinter allows you to mine on their pool with OpenCL miner through Java Webstart.
www.coined.com has undefined status, WOT reports it as an abusive site.
WebCL bitcoin miner looks like a fully functional WebCL miner
Bitcoin Plus allows one to mine through a Java Applet, but it is a (highly inefficient) CPU miner.
And there are a couple other less notable web CPU miners.
There is a page claiming to be an OpenCL miner at coined.com but I'm not sure if it's functional or not.
Would http://webcl.nokiaresearch.com/jsoclbm/ meet your needs?
Here's a current option for adding mining to your website: https://coinhive.com/