Can a first-time buyer that is willing to live in one of the units in a 4-unit MFR still house hack if he partnered with an LLC as part owner? If so, what type of financing would be available? Thank You

Once an LLC gets involved, most of the time you will have to go commercial lending with 20% or more down. It will really depend a lot on your state and local lenders. But that's how almost all of the lenders around me work once an LLC gets involved.

LLCs involve commercial lending.  You can still house hack, but the terms aren't as good.  Just make the numbers work.