Without putting compiled code on the user's machine, then more than likely, no. It would require you to specify to the os a specific program to run, and that's going to violate the security restrictions that are (and should) be on most browsers in the market today.
You might be able to create plug-ins for each of the major browser vendors which will intercept the link and open it in IE, but that becomes tedious, as there are different models for each browser. On top of that, you have to get your plugin installed on each machine, which users might not be interested in doing.
The best option here is to inform the user the site must be browsed in IE, and if possible, work with the vendor to make it run in other browsers.
Another possibility is that you might host both your app and the vendor app in a client side program which uses an embedded WebBrowser control, which is essentially, IE.