One law blogger put up this list of solutions, well worth looking into. I looked at the offerings, and Bill4Time is the one that offers a) logins for multiple users, and b) a way to integrate with QuickBooks, which would be crucial for your operation. At $30/month, it looks pretty tight.
That blogger had a few articles on web-based practice management, and seems to have concluded that Rocket Matter and Clio are the two most suited for a smaller firm. I've had one client try out Clio, but it didn't work for them. (Worth mentioning that they offer a discount through ABA.
A law blog that gets frequently referenced in my world is The Mac Lawyer, who really loves him some Rocket Matter. The service is pricey — y'all would pay 60+(50*5)+(6*40)=$550/month, and since it's unclear to me that you would use all of its features, Rocket Matter may not be the direction you want to head. (Though I have to say that, if I were starting a law firm today, I'd sign up for Rocket Matter straight away.)
Not specific to the legal industry:
Whenever I want to search for "cloud-based" solutions, I start with Google Apps Marketplace, not because it's comprehensive, but because I can cover a lot of ground quickly, getting a sense of the current offerings with some very helpful user feedback.
In their Accounting section, I found a couple of services that look very promising: I like FreeAgent's pricing scheme of $20/month with no restrictions. And Invoicera received some good reviews. I have looked at the Zoho products before, and while they seem nice and clean, I always run into some major thing that they're missing. If my new switch to Xero for online accounting doesn't pan out, I might give FreeAgent a go.