There is presently no state-wide vaccine requirement for public spaces that I can find in Washington, but King County (which includes Seattle) does have a requirement to show proof of vaccination to enter restaurants, bars, and certain indoor or large outdoor recreational events. According to the page explaining the requirement, acceptable forms of proof include:

• A CDC vaccine card or photo of vaccine card.
• Documented proof of vaccination from medical record or vaccine provider.
• Proof of vaccination from another state or county.
• Printed certificate or digital record from MyIRMobile.com or other apps.
• QR Code or digital copy of your COVID-19 vaccination record from WA Verify

I believe that "county" above is supposed to say "country"; this typo is fixed on the kcvaxverified website, and the implementation guidance for businesses linked there explicitly states (in large type on the bottom of page 8)

Note: Proof of vaccination from another country is also valid.

Your mother is not eligible for a CDC card because (you say) she was not vaccinated in the US, so she should be prepared to show her Uzbek proof of vaccination, probably accompanied by an English translation if the proof she has does not already include English.

It is likely that many businesses will not have seen an Uzbek proof of vaccination before, but most businesses will want you as a customer, so be prepared to be patient and polite, and you should usually be successful.

There is presently no state-wide vaccine requirement for public spaces that I can find in Washington, but King County (which includes Seattle) does have a requirement to show proof of vaccination to enter restaurants, bars, and certain indoor or large outdoor recreational events.

The page above talks about proof from "another county", but this is a typo, which is fixed on the kcvaxverified website, and the implementation guidance for businesses linked there explicitly states (in large type on the bottom of page 8)

Note: Proof of vaccination from another country is also valid.

The official order from King County states that (emphasis added):

Verification of vaccination may be established by any of the following methods of documentation:

• CDC COVID-19 Vaccine Record Card, or photo of CDC Vaccine Record Card
• QR Code, digital or printed certificate from MyIRmobile.com (WA Department of Health partner website, My Immunization Records)
• Vaccine administration record from a vaccine provider, including an individual’s doctor, pharmacy, or other official immunization record from within or outside the U.S., including a photo or photocopy
• Specific mobile applications that verify vaccination status will be allowed if approved by the WA Department of Health or Public Health – Seattle & King County

It goes on to say that (emphasis in original)

Employers, their employees, and volunteers, must implement this ORDER equitably and may not:

• Scrutinize proof of vaccination more closely based on an individual’s race, national origin, religion, sexual orientation, gender identity, disability, age, or
• Refuse to accept valid proof of vaccination that is authorized by this ORDER, such as WHO approved vaccination from countries outside of the U.S. or photographs of CDC vaccination cards.

Your mother is not eligible for a CDC card because (you say) she was not vaccinated in the US, so she should be prepared to show her Uzbek proof of vaccination, probably accompanied by an English translation if the proof she has does not already include English.

It is likely that many businesses will not have seen an Uzbek proof of vaccination before, but most businesses will want you as a customer, so be prepared to be patient and polite, and you should usually be successful.