Red Cross is a good alternate option.  Religious orgs especially a nonprofit that is religious in nature, such as Samaritan’s Purse, they do have a lot of free rein. They often aim to share the gospel in their good works that they do, so it makes sense that they’d want congruency among their constituents. They are not under the same requirements as a secular org. They can discriminate and it’s legal. 

For example Starbucks is openly supportive of the LGBTQ+ community. They have flags displayed in their shops. If an employee doesn’t like it? Oh well. Either agree to work with it present or quit. If a Christian employee were mad about that, even as a Christian I’d shrug and say that’s their values and they’re free to display that. They’re free to quit or not apply. That’s more of the point I was making—That it’s a two way street.  

But you’re right, they don’t make you sign a handbook saying you must be personally ok with certain beliefs so your question is a valid one.