>> How can I send email out of my web apps using the clients own Microsoft outlook
you can't. And you don't want to.
You can't because the browser (which is all the software they have) cannot interact with other programs on the computer. You could install a software program on that client computer which acts as a link between the browser and the email client, but you REALLY don't want to do that.
You don't want to send via their Outlook client anyway, because that would make your program useless to anyone not running Outlook, or running on a device which doesn't have Outlook.
Clearly a better solution to your actual problem is desirable.
>> The client wants to use their own signatures
So allow them to create signatures in your program, and attach them where they are appropriate.
cheers
Bruce