You need your password login user name for ti sign to the App Store.
See the following Apple articles:
If you forgot your Mac login password -> https://support.apple.com/en-us/HT202860
Change the login password on Mac -> https://support.apple.com/en-gb/guide/mac-help/mchlp1550/mac
Reset your Mac login password -> https://support.apple.com/en-mn/guide/mac-help/mh35902/mac
If you can't reset your Mac login password -> https://support.apple.com/en-gb/HT212190
See also -> Ways to remember passwords on Mac
Ways to remember passwords on Mac – Apple Support (UK)
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You are sign in here . . . with your Apple ID & Password.
Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.
You can’t merge multiple Apple ID accounts.
What happens when you use the same Apple ID & Password that you use to sign in here?
See -> Sign in with your Apple ID -> Sign in with your Apple ID - Apple Support
Manage and use your Apple ID -> Manage and use your Apple ID – Apple Support (UK)
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I have a mac running 10.9.5
Your Mac is on macOS Mavericks.
If there is an update for your Mac you will see in the Dock:
If your Mac is eligible for an update, check here -> How to download macOS -> How to download macOS - Apple Support