You cannot download LE8 from the App Store and all you can download from Apple's website are the updates, not the full program, which is only available on DVD.
You should be able to transfer LE8 from your old to your new MacBook with a firewire cable, ethernet cable, or a wireless network using either Migration Assistant or Target Disk Mode. Check out these two videos:
You can Google these if you need more information. I hope this helps you get up and running again on your new MacBook.
Target Disk Mode would allow you to select only what you want to transfer. In addition to the application itself:
Alternatively, make a disk image of the Installer dvd on the old Mac, copy/move it to the new one and use the image to install,
Use the Remote Disk function