Can Kazakh nationals apply for UK visas and travel to the country?

It will be very difficult or perhaps impossible for her to secure a UK Standard Visitor's Visa. The reason is: she's your wife.

A successful visa application convinces UKVI that the applicant will leave the UK before visa expiration, presumably to return to the home country. The reasons for returning to the home country are commonly called "strong ties." The usual things presented are family members (parents, children, spouse, etc.) remaining in the home country; a good and permanent job where the employer agrees in writing that the applicant may take this time off to travel and will have the job waiting upon return; ownership of a home or business in the home country; and anything else that would draw the applicant to return to the home country, rather than overstaying in the UK.

Use the search box at the top of any "Travel.SE" page, and you can easily find many questions and answers addressing "Strong ties" and "return to home country."

Even if strong ties exist and are presented in the application, I am not convinced UKVI will allow her to visit. UKVI will assess that a married couple will or should or is invariably aiming at living together. They will believe that the real agenda is to come to the UK and live there with you. In their view, your marriage weighs heavily in increasing the liklihood she will overstay.

All in all, I think her chances of securing a UK visa are very slight. Your best course of action would be, before a visa application is filed, to consult in the UK with a solicitor well-versed in visas and immigration. A solicitor can advise you how to best present the info, or, perhaps, suggest another course of action if a visitor visa is truly not in the cards.

Finally: this Answer does not address Covid-related travel issues, which may also affect travel to and entry into the UK.

The UK borders aren't closed to foreigners (except if having spent the past 10 days in certain countries, NOT including Kazakhstan), so she's perfectly free to travel. Only the usual visa conditions need to be fulfilled, and then registering at https://www.gov.uk/provide-journey-contact-details-before-travel-uk 48 hours before departure AND having a negative PCR, LAMP or Antigen test taken max 3 days before departure.