**Short answer**
People with HIV can get tattoos.
**Background**
In Africa there are countries that tattoo people identified with HIV (Source: Kenya Today) and some people with HIV find comfort in tattooing biohazard symbols and related images on themselves to express their illness (Source: CNN).
However, as rightly mentioned by @AMR, macrophages which are mainly responsible for ink fixation express CD4 and therefore can be infected by the HIV virus targets.
Hence, while I don't think HIV affects tattooing given the fact that HIV-infected people get tattoos, in theory HIV could reduce the ink fixation.