I would fix the FF bugs ASAP. You'll have to do it eventually so put that at the top of your to-do list. The fixes probably aren't that hard. Thanks to firebug you could probably pin-point the bugs really fast.
It's probably better to put off the launch for a few days to avoid making a really bad impression to potential users.
What you can take from this is while being IE compatible only will give you about 58% of users, you can see that FF is growing steadily and will probably catch up someday in the not too distant future.
It also depends on what kind of web app this is. If it's something for younger people, or people more technology inclined, the FF visits may be a lot higher than the graph suggests.
I feel for you. When I wrote my first AJAX program, there was a lot of javascript controlling the UI. It was almost complete before I looked at it in IE for the first time. Two days and 10,000 alert boxes later I had it working.
It's a necessary evil. No one enjoys it, but if you don't make the fixes you'll prevent a lot of people (including myself, a FF user) from ever even trying your website. If I try it once and it completely breaks, I'm never going to come back.
If there are one or two non-critical features that are creating all of the bugs, then perhaps remove them initially and add them once you've fixed the compatibility bugs.
Did you use a library such as open rico or something based on prototype? If not, consider porting your existing code to use them as those libraries have solved many of the complexities of writing cross browser AJAX code.
Also, if you don't have this book yet, get it now. It lists all javascript and css methods and which versions of which browsers support them. I relied on it heavily when writing/debugging my web app user interface.
My app is close to ready to be launched, but it only works in IE. Is it ok to launch with it working in IE only? Do you think I should let users know that it only works in IE?
As someone who doesn't use IE, I'm more impressed by a site saying "This site is only compatible with IE right now - we're working on it, though!" than with a site that is obviously broken and incompatible. And if you don't intend to ever make it work in the other browsers, say so up front.
And users will figure out very quickly that it only works in IE; better to be up-front about it.
Definitely let users know, and block access from other browsers. It's better to see an "Sorry, IE-only for now, please come back later" message than a broken web app.
Personally, I can't imagine starting up IE just to check out a new web app, unless it seems insanely cool... so if you're looking at not alienating FF users, it's a good idea to get it working in FF asap (Firebug makes JS coding sooo nice).
One thing to remember is if it works in Firefox, it works on every platform. Windows people can still use your site even if they need to click another icon first...
IE-only means you lose all the Mac people, all the Linux people, and probably turn off the majority of the more clued-in Windows people.
Early adopters -- your target audience I assume -- don't tend to use IE.
It's probably better to put off the launch for a few days to avoid making a really bad impression to potential users.