H1-B applicants at non-profits

I'm looking into non-profit jobs in Washington D.C. (or maybe Chicago), but I would need a H1-B visa for what would most certainly be an entry-level job. My hope with non-profits is that some of them are cap exempt and more willing to sponsor a visa than other employers. I'm also aware that they are probably stretched thin in terms of resources.

I did my undergrad in the US at a prestigious institution with a major in the humanities. I'm currently going into a master's program in human rights in Europe. The program is also accredited in the US. I'm European and lead a national intersectional anti-racist organization in my home country. The non-profits I'm looking into deal with human rights, anti-racism and promoting democracy. I recognize I'm searching rather broadly.

I was hoping for advice from professionals in the field to see if hiring on a H1-B visa is at all common practice. I'm slightly reluctant to approach these organizations with this question as a way of making my first contact with them.

Similarly, I see non-profits posting remote volunteering opportunities. Are they generally open to someone not on a some kind of visa already? Any advice on either of these issues is very much appreciated.

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