HRC has a petition supporting the Equality Act, link following a major news story about its introduction. Today 31 states lack civil rights protection on sexual orientation and gender identity issues.
As for the Executive Order regarding “religious freedom”, here’s an older letter on the topic sent to Trump on February 1 HRC has a petition on the matter here. Politico reports Trump will sign the XO Thursday. Would the order apply to religious groups that receive federal funds? That’s the scary part.
On the refugee issue, most of the mainstream faith-based social services agencies (based on Protestant or Catholic churches) seem to be fine in handling LGBT refugees. The administration could argue that some narrower religious groups do help provide services to their own people and must be allowed the freedom. In practice, the order is very unlikely to affect GLBT Americans.
There have been a few issues though with a few charities. One local church broke ties with World Vision in supervising youth group “famines” for hunger because of reported ties to anit-gay efforts in Africa.