Since you are asking about sit-ups, I am assuming you wish to develop your abdominals or core muscles. Core muscles are the muscles that make up your entire torso plus your gluts and shoulders. Sit-ups are not that effective of an exercise for the core, so there are plenty of things you can do instead. Exercises that have you stationary in a plank, either in a prone position or a supine position will strengthen the core including the ab muscles. Since you are constricting the muscles in an isometric way and not moving, you should not experience vertigo. For an alternative to sit-ups, try the Roman chair or hanging rings. The Roman chair is done by supporting yourself in an upright position with the forearms and bringing the legs or knees to the abs. Hanging rings are suspended from the ceiling to provide support. The plank can be done in a prone position from the rings or the same movement mimicking the Roman chair can be done using a rolling platform, towels or gliders on the floor.
I would recommend investing in yourself and asking a personal trainer to work with you for just one or two sessions so you can do these exercises safely and correctly.