I assume you are asking about New York City? If so, you should know that NYC is unique and most of the information from out-of-state responders could be incorrect.
Yes, to purchase a cooperative you take out a loan just as you would for any other property. Wells Fargo, Citibank, Chase, etc. etc. all provide loans for this purpose--all the time, every day. This is totally routine in the New York City area.
Co-op income requirements are imposed not by the lenders, but by the co-op boards. These are residents who govern the building. They have great leeway in setting up requirements. They can require a certain down payment percentage, or require all-cash purchasing only, or require that you have $X in assets. They can also reject people because of celebrity status; Richard Nixon was famously rejected by a co-op near Central Park, and there are other examples that surface in the gossip pages from time to time.
You may be asking about co-op units that are relatively rare, and that are found on the Lower East Side and Harlem, in my experience. The unit could have a selling price of $400,000+, say, and a family income limit of $90,000 imposed by the co-op board.
These income restrictions/limits are usually a legacy of how the property was developed, the philosophy and incentives in place at that time, and they tend to go away with time. Usually the idea was to prevent rich people from buying into the co-op, sort of the reverse of the situation you'll fine in the most exclusive co-ops near Central Park.
Unfortunately, to buy in such a unit, a buyer may need to have a lot of cash on hand for an unusually large down payment, and then the buyer can get a loan as described above. Any restriction as to who can buy into a building tends to hold down selling prices.
Licensed Real Estate Salesperson
Charles Rutenberg Realty, LLC
New York, NY 10021