This would be a good time to reiterate the official link on “obtaining asylum in the United States” especially for LGBTQ people from hostile countries (like Russia, Nigeria).
The DC Center is affiliated with “Center Global” which does maintain a list of “asylees” in need. This group has not gotten much consistent coverage from the major media. But the clients here seem to be people already in the country, settled once and apparently needed more help again. That’s a different process from settling refugees through established charities (Catholic Charities, Lutheran Social Services) who work with the federal government closely on carefully screened refugees, before the refugees arrive. Most of these individuals probably arrived through a variety of particular circumstances outside the well-structured programs run by social service agencies with the government, and often with the help of larger church congregations.
This is not a process where people overseas wait for a “sponsor” to come here. Outside of foreign adoption of children, that would be “private sponsorship” of “new” refugees which is not legal in the US now (given the political climate), although it is done in Canada.
During the Mariel Boatlife from Cuba in 1980s, there were many LGBTQ Cuban refugees already in the US (mostly in southern states. Especially Florida), looking for hosts with “spare bedrooms”. They were "refugees" who had arrived through urgent circumstances that normally would have made them illegal (as if they had crossed from Mexico). That sort of issue could develop gradually with previously settled refugees, but it has gotten very little coverage so far in the U.S. during the recent crises involving Syria, much of Africa, and even Russia.
It’s possible that people will be contacted by “friends” especially on Facebook looking for assistance overseas. This is a matter that requires great caution, and not much has been written about it yet. In some cases, people overseas may mistakenly believe that “friending” someone gives them better status in refugee application process; there is some evidence that this is going on.
Update: Aug. 6
It's been explained to me that some asylees are in the US on visas (like student visas) which could have expired. A member of a marginalized group (meeting legal standards) can file for asylum, In a few cases, undocumented people (or people whose visas have just expired) can file for asylum and remain here legally while asylum is processed. That could make helping asylees more personally challenging that refugees, who must go through one of the large approved social service agencies first. Asylees have not necessarily always had the support of one of these agencies, They could have come from stable countries on student or employment visas, and fear going back because of political changes in their home countries (like passages of anti-gay laws while they are already here).