I have old MAC and I am limited to OS X 10.7.5 and Safari browser 6.1.6. Web Sites access issues

What makes you think you cannot upgrade to a newer OS.

The App Store Updates page only informs you of updates available for the OS you are running.

Installing a newer OS is called an upgrade.

Which OS you can upgrade to will be governed by which model and year your mac is.

Click on the  top left of your screen click on About This Mac to find out the year of your mac.

For older OS’s,

Open Safari and click on this link, 

How to get old versions of macOS – Apple Support

Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.

Then go to the Download OS section and click on the OS you want to install.

If you want to or are able to install Big Sur, open Safari and click on the link below.

How to upgrade to macOS Big Sur - Apple Support

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see

if they are providing 64bit updates or alternatives.

If you downgrade to OSX 10.6 (must perform a clean install by first erasing the drive) it may be possible to run TenFourFox which is a specially backported version of the current Firefox (made for PPC systems, but OSX 10.6 is able to run Rosetta so this combination may work).

Another option would be to install Linux on the old Mac which would give a current non-Apple OS capable of using the current versions of the popluar web browsers such as Firefox, Google Chrome, or Vivaldi. This will require you to learn a new OS. As long as you don't have a first generation Intel Mac, then this would be a good option. Linux Mint or one of the Ubuntu flavors such as Kubuntu or Ubuntu MATE are good choices as they can run well on older systems with limited RAM (4GB is best, but it is possible to run with 3GB or possibly even 2GB). You can even test it out to see what it looks like by using the "demo" mode on the USB installer. Just make sure to have a good backup and a way of reinstalling or restoring macOS if you change your mind after installing Linux.

Sorry you are having these problems. Not to sound preachy, but these are the problems we run into when we no longer upgrade (voluntarily or otherwise). So, yes, you may have to use different browsers for different sites. Another one to try is Oprah. I'm sure there are others out here as well. Here is some information that may be useful to know. As you may know, Safari, is installed as part of the operating system, normally it cannot be individually upgraded. it is periodically updated when the computer's operating system is updated or upgraded. Web site engineers write their software to work with specific browsers. For some reason, they often choose to not be compliant with Safari (or some versions of Safari). So in summary, different versions of different browsers may be compliant and will work with different web sites. Just as browsers are updated periodically, so are web sites.