We have a concrete stairway that winds up to the house from the street. There are several landings. The first two steps at the sidewalk have sunk and are sloping. How difficult is it to repair this without having to replace the entire landing? And what type of contractor should we get? Limited budget... trying to sell the house and need to make cost-effective repairs.

The concrete is most likely broken in the area of concrete behind the back corner of the second step. Sloped steps can be extremely dangerous for obvious reasons. Proper repair is very difficult. You may find a handyman that is willing to try and jack it up and stabilize but probably not anything most concrete contractors would be willing to get involved in. You may be able to ask around and find some carpenter or cement worker willing to take it on. I would not because I don't see a way to tie it back together-no access probably.
My concern would be the stability off the step system above the broken step also.
But you may find someone who can find a way to stabilize it. Home improvement big box stores usually have bulletin boards where some tradesmen and handyman types may leave a flyer or card.
I would want to disclose that repair in whatever fashion is required or customary in your area.
The buyers should have an opportunity to have it inspected by a home inspector of their choosing. Getting this disclosure documented in writing may save you some serious problems in the future, depending on the laws in your area.
Without seeing it, I cannot say that it is a critcal items or not but it sounds like it could be.