I am currently a graduate student at a research institution (not a university). My stipend was paid based on my professor's grants, but it seems something happened with his grants that he told me he could not further support my PhD study. Some other labs I am interested in in the same department cannot take me as well for the same reason. And it is a research institution so I cannot be on TA for stipend. Is it possible to apply for another graduate program, since many programs have prejudices towards students reapplying? My advisor agreed to write a letter, but I am not sure if he will talk about the research funding since it is a sensitive topic.
I doubt you will find any prejudice on that issue. But you need to meet all expectations and requirements for any program you apply to. The advisor does need to say that you have no funding available due to the nature of your current position, but doesn't need to say more than that.
But if the only issue is the funding, then that needs to be brought out so that people don't make assumptions about your ability.
If your professor has a wide circle of contacts, especially collaborators, they can, perhaps, get you special consideration with one of those people. That sort of assurance of the competence of a student can go a long way.
He has the responsibility here. Don't let him off too easy.