Sorry to hear you are having these issues. It seems to be that quite a few folks have frustrating experiences with the App Store and Mac OS upgrading. I myself have been fortunate, in that I always make a copy of the "Install macOS whatever" file I download from the App store. However, I still "miffed" that within my Purchases tab, Sierra (OS 10.12) is not listed there! (I have the last "Install macOS Sierra" file, although I am running High Sierra, OS 10.13.3, on both of my Macs).
First question is, what exact Mac model do you have?
Secondly, if you do upgrade to a new Mac OS, you'll need to insure that all your third party (ie, non-Apple) software is compatible with the new OS you are upgrading to.
Third, it was my understanding that once you have Snow Leopard (OS 10.6.8), you would be able to "see" at least OS 10.7, Lion, in the App Store, and upgrade to it. When you visit the App Store and click on the Featured tab, what Mac OS do you see there?
I wish you lived near me, as within my Purchased tab, I can see Lion, Mavericks, Yosemite, and El Capitan, and thus I could provide you a copy of any of those.
One other thing: upgrading from OS 10.6.x to El Capitan (OS 10.11.6) is a huge leap. You'll need to make all the necessary preparations (disk cleanup/maintenance/repairs, backups, software upgrades, etc.) before proceeding.