The definition of a domestic partner varies according to the state where they live.
Before granting domestic partner insurance coverage, the company may require proof that the applicants meet the standard to be considered domestic partners.
If you are interested in getting domestic partner insurance and your employer offers a benefits package, you can start by contacting the plan administrator to ask if your coverage extends to your partner as well.
In a case where a domestic partner is recognized under an employer health plan, this is probably the easiest way to proceed.
There are private health insurance companies that will offer domestic partner coverage and you can search for them in the same way that you look for individual insurance.