If atoms are mostly empty space then how are we able to interact with things?
When we think about the structure of atoms and molecules we need to use a range of simplifying analogies as the objects we are talking about are too small to see and behave according to quantum rules we are unfamiliar with in the macroscopic world. None of the analogies we use are perfect and may mislead unless they are brought back to reality by recognising their limitations an checking their implications again the mathematics we know that does describe their behaviour.

That is the problem here. On one analogy atoms are built from nuclei containing protons and neutrons surrounded by electrons orbiting a long way away. This analogy is useful for explaining what happens when, for example, alpha particles are fired at a gold film a few layers of atoms thick. This is the famous Rutherford scattering experiment first done in 1909. This experiment led to the "solar system" analogy for what atoms look like. But really it only explains what they look like from the point of view of fast alpha particles. The mass of atoms is concentrated in the small nucleus and it is the nucleus that alpha particles bounce off, occasionally.

But that is only one, imperfect, way to describe what atoms and molecules look like. It doesn't describe how individual atoms join to form molecules, for example. For that we have to look at how electrons behave. Here a different range of analogies or mathematical descriptions are needed. The solar-system view of electron behaviour is deeply inadequate to describe what electrons actually do. And it is a very poor description of where electrons are. It is better to think of electrons as occupying 'clouds' of probability than being at a single point in space. Those clouds have a range of shapes from simple spheres to multi-lobed, multi-layer things in atoms with many electrons. Many of these have some probability of the electron being anywhere inside the cloud even, in some cases, right into the nucleus. In this analogy atoms are not empty at all but are filled with some probabilistic amounts of electron. No more than two electrons can occupy any orbital which imposes some strong constraints on what can happen when the clouds of two atoms or molecules meet.

Almost all the interactions we experience in normal macroscopic life involve those clouds of electrons interacting with each other. There are a range of different ways they can do this, but at the heart of things they all consist of electrostatic interactions at the atomic or molecular level where electron clouds push (or pull) each other. The low density of electron clouds doesn't matter since the quantum mechanical rules determine what can happen when the clouds meet. Normal interactions in the real world are just the macroscopic impact of these interactions.