can I update my macOS Sierra version 10.12.6 to macOS version 10.3?

I am assuming that you mean 10.13 which is macOS High Sierra.

All macs running macOS Sierra 10.12 can upgrade to macOS High Sierra.

As noted above your mac may be able to upgrade further depending on its year of manufacture.

Please use Safari to access the following links do not use Firefox

How to upgrade to macOS Mojave - Apple Support

How to upgrade to macOS High Sierra - 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 be warned that it does 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/. Once the list has been created you can then

contact the developers of the apps to see if they provide 64bit updates. 

How to upgrade to macOS Catalina – Apple Support

Your question doesn't make sense. OSX 10.3 is ancient, and not an upgrade, so I assume you've mis-typed. In any case, what you can upgrade to is primarily dependent on what model of Mac you have.

To check that, go to 'About This Mac' from the Apple menu, click 'System Report' to open System Information (which is also in Applications/Utilities), and the display will show you your Mac model with a name and two numbers separated by a comma, for example iMac19,1 (a recent Mac).

Post that, and we can work out what you can upgrade to.