But normally, the appraisal is owned by the old lender  (they are the client to the appraisal company), so you will need their permission.

Was wondering if this is normal or how to go about finding a right property manager who sees the true value? 

I tried one of the 5 the normal way with everything almost funded but it was a huge pain with the drawers (they deposited with checks for some reason so it took an additional few days to get the funds) and I had to deal with a super lame system to even get the drawers sent over.

By the end of 2020, lenders and banks had learned their lesson, and stopped trying to compete on that basis (lo and behold,  once everyone stopped trying to be a rate whore, Q4 wound up being the most profitable quarter ever, for the mortgage industry, and Q4 is normally the slow season!).

Funding a new project with 75% LTV from an existing property is normal.

I was intending on making an offer on a 4plex with a sale price of 175,000The company that is selling the property is RealPro in Fort Lauderdale.They are asking for a 10% deposit with offer - is that a normal amount?

Most lenders are hesitant to give you financing with no skin in the game for the first deal. 

I think @Dmitriy Fomichenko or @Brian Eastman might have a list of lenders who SPECIALIZE in this type of loan, which is different than a 'normal' rental loan.Dan Dietz

We don't want things to get too normal and easy.

You could go to your normal bank to apply but I would recommend you find a smaller lender who is able to work with your personally.