I had this same problem. I installed 10.6.3 on an old iMac late 2006 a1207 which can only go up to 10.7.5 Lion.
After installing Snow Leopard 10.6.3 with the disk I was unable to update the software through the updater in OS X. I would get stuck during restart with only a mouse. Also, once I did manage to update to 10.6.8 (where the App Store comes available), I was unable to click the download button in the App Store since it would not do anything.
These are the steps in order that I followed to update Snow Leopard to a state where I was able to download Lion from the App Store thereby enabling myself to install that and create the recovery partition:
- Mac OS X 10.6.8 Update Combo v1.1
- Download Mac OS X 10.6.8 Update v.1.1
- Update software through the OS X updater (click the Apple logo left hand upper screen > Software Update...)
And yes, this sounds weird since both those updates have been posted by Apple on the same day and should (I suspect) contain the same updates.
The first update gave me access to the App Store but then I got the problem with not being able to click the download button.
The second update fixed my issue with my updates getting stuck so then as a third step I was able to update the software through the OS X updater.
After the third step I was also able to click the download button in the App Store.
That is how I finally managed to download Lion from the App Store and install that together with the recovery partition (which comes with Lion) on my mac. Old Macs are definitely a hobby considering the amount of time they take ;).
NOTE: While Mountain Lion as higher will be available for download in the App Store, for Lion you must have bought a copy from Apple https://www.apple.com/shop/product/D6106Z/A/os-x-lion.
NOTE: While you are at it, it might be worthwhile to download all previous OS X version you can from the App Store and burn them on a dual layer dvd or create bootable usb sticks since after updating to Mojave 10.14.x you won't have access to those downloads anymore through the App Store (only through the recovery partition of a Mac and also then you may only download the version currently present on the machine and upgrade from there). So no more downgrading without physical installers/cd's.