I agree with some of the answers where it says most of the requirement to use a specific bank for pre-approval are probably properties that are lender owned.
They do usually have a specifid mortage broker whom they are familiar with and comfortable to work with (there are different quality of mortgage brokers and leners out there, the REO lender wants to makes ure you are preappoved by one that they trust). They usually do not require you to use that lender (that would violate the RESPA rules of not steering); but will allow you to have free appraisal if you do use that specific lender.
Otherwise, from Lender Owned properties.
Getting yourselves preapproved is always good. After you are preapproved by the selected bank, and if your offer is accepted, you are not, however, obliged to use that lender, and you can choose wnomever you want.
The legality is in the laws of the state you are in. In NC they cannot require that you finance from a specific bank. Many sellers and builders will offer incentives to get you to go with a preferred lender. By no means would they require that you use a specific lender that would cut down on your potential buyers. So I believe they would be shooting themselves in the foot. Of course they maybe are just wanting you to pre-approve with this institution and they are going to charge you to pull your credit and run the approval. That is only speculation. Without more facts I cannot say.
By law, it is illegal for us as Real Estate professionals to give legal advice. We can however give our professional oppinion as long as we don't cross the line. In as best to not cross that line and help with your question, if a seller requests that an agent receive a pre-approval offer from any perspective buyers, we are bound to obey them as long as it doesn't require us to break the law. In as well if that same seller requests from us to provide a pre-approval from a certain bank / their bank, then we would be required to request that from the perspective buyers. That in no way should bind that buyer to that lender, unless the buyer agreed to this in writing prior to making that application. Hope this helps.
I don't think there is any legal basis why a Seller or Agent couldn't ask for a pre-approval from a particular lender. I have seen that in the past.
It is possible that they previously had a problem with receiving "bogus" pre-approvals and want to have someone they trust verify that the Buyers are truly qualified. Now, I don't think that they can force the Buyer to actually use that lender to purchase the property.
Hopefully this answers your question