Yes, you can apply for H1B visas using Caribbean passports obtained through citizenship by investment programs. There are no restrictions.

The H-1B is a temporary (non-immigrant) visa category allowing employers to hire highly skilled professionals to work in the US.

There are close to 3 million H1B visa holders in the U.S. The H-1B visa program allows companies and other employers in the United States to temporarily employ foreign workers in occupations that require the theoretical and practical application of a body of highly specialized knowledge and a bachelor’s degree or higher in the specific specialty, or its equivalent. Congress set the current annual regular cap for the H-1B category at 65,000 limit. There are 20,000 additional visas for foreign professionals who graduate with a master’s degree or

Let us take a look at the statistics provided by US authorities for Caribbean in 2021

Country (POB)H-1B visas (2021)
Antigua and Barbuda17
St Kitts and Nevis7
Saint Lucia30

The U.S. Embassy in Barbados has resumed limited routine visa services effective March 1, 2022. If you have alternate citizenship or residency, you can also apply for visa services at the nearest U.S embassy where you live.

Source: USCIS