@Anthony To I'm from Arizona and worked for a few years at Wells Fargo as a loan officer. Most small credit unions and large regional banks should be able to underwrite a loan for 50k without any issues. Like most other's indicated you'll want/need to put down 20% on a SFH and 25% on a Multi-family.
Some lenders or loan officers will flat out lie to you and say they can't do a loan under 100k or 50k or whatever number they decide to tell you. This is for a couple reasons, the first being they simply make a lot less commission on a home that's 50k compared to a home that's 250k or 500k and they both require the same effort. The second reason is because what happens sometimes is they will get half way through the loan and closing cost will come back higher than anticipated and now the loan is dead. But you as the buyer have already paid for an appraisal and credit report that are non refundable. Not the ideal situation to put yourself in if your a loan officer.
In Arizona and most states in the US their is a maximum closing cost percentage of the loan amount. So let's say your buying a home for 50k and the maximum closing cost percentage is 7% of the loan amount. That would equal $3,500. So if all of your closing cost combined are greater than $3,500 the lender can NOT legally underwrite the loan. After you pay $650 for an appraisal, $1,000 for property taxes, $1,000 for origination, $650 for title and so on it's very easy to get close or pass that magic number. I don't recall what the maximum is off the top of my head so 7% is just an example. But I hope that sheds some light on to why and how the bank actually looks at the loan. One other thing is often times the lender will make you pay points for a lower rate because the loan is so small and this is one more thing that can contribute to closing cost being inflated.
Most of AZ loans around $50k will go through but as you look back east because of property taxes it is more challenging to make them happen. Good luck and let me know if you need any help with running the numbers.