You can try out any of the Backup App, that would let you have fun!
The thing that you're saying
I recently lost all the emails on my laptop computer. And they were important business emails. Many of those emails still exist on my Android phone.
As the mails on your device might get removed as soon as you connect it to the Internet (sync might do that), if you need to install a backup app better download its
.apk file to your computer, and sideload it (e.g. copy it to your SDCard, and start the
.apk using any file manager, which would trigger the installation; make sure to have unknown-sources activated for this). A good choice would be Helium Backup: it creates
.ab (Android Backup) files compatible with
adb backup, and there are solutions available to extract those on your PC.
Is there any way to pull the emails off and then drop them in to my mail program?
No, there is no direct way to do that – you could only forward them, which you don't want. Then I don't think there is any other way to do this. You can either create a backup file to just save the emails, then forward them. That's the only way to get back your emails, but still you will lose the time, sender name and other! And it will be replaced with Sender as: You, Time: Current Time, Body: Body will have the email included as signature or something like that. That's it!
Good luck, Cheers.