The mortgage brokers will likely have access to more funding sources and can probably get more creative with the financing than a local bank or credit union.Can you provide more details on your situation? 

My understanding is that if you have funds available for transaction A-B (hard money, transactional,etc), you would be able to do transaction B-C because you have already purchased the property with the funding source.

You have come to right place for real estate investing sources.

I would continue to do this until I was comfortable buying a house solely to be my primary residence for a long time.

Sources Sought has closed a very long time ago. 

Is the practice that the banks and hedge funds use open to LLC's, partnerships, sole proprietors,etc?

You might want to look up @Dave Van Horn as a source for NPN product. 

I've currently been investing in REO's in the Chicagoland area for the past few years, primarily flips but it seems like a lot of the "good deals" are stating to go away so I'm looking at different sources now.  

I just want to find 1 source with the details so I can start.

You may have other income sources or access to people who can co-sign a loan etc., but based on the numbers you provided, I predict you will have a VERY hard time qualifying for a big enough mortgage for a solid property in a solid neighborhood, let alone two.