Apple's download servers have some pretty old stuff, you can likely auto-update back to OS X 10.2.8, if you have a need for that. A 10.6.8 system has two methods for updates -- the App Store, but better would be Software Update, under the Apple Menu.
I think more likely that you simply don't have any updates available, and your Snow Leopard system is fully updated. The only likely "update" that you would have is to upgrade the OS. And, the only update that would be offered would be High Sierra. The banner for that is probably visible on your App Store Updates tab.
I would NOT do that upgrade without 4 GB RAM or more, and an SSD installed, and I think the upgrade won't work anyway, unless you upgrade to something in between first, like OS X 10.9 (Mavericks). Those systems installers between 10.6 and 10.13 are all available, already on USB thumb drives, often on
eBay somewhere.
Easier (and cheaper) is to download the macOS installer app of your choice, and use that to create a bootable installer. There's lots of places that offer tips on how to do that.
One other tip that I would have for you: Go to Apple support downloads page, and search for 10.6.8 Combined.
You should get
this page. Download that combo updater, and run it on your MacBook.
That is a tip that is long known to help with some kinds of system problems, as it will clean up caches, and assure that all components in the system are both installed, and installed in the correct locations. That may help enough so that your system will then work as you would hope.
(and you should make sure that time and date are set correctly, as that simple thing can also be a fix for certain problems, like connecting to remote servers (such as Apple's

)
If you want a web browser that is a bit more modern than Safari in Snow Leopard, and one that may work more reliably for you,
download Roccat web browser.