If an atom is the smallest particle in the universe, does that mean that the spaces in between the atoms in water when being heated (expansion) are vacuum?
Atoms are not the smallest particles. Atoms are made up of a nucleus (consisting of protons & neutrons) surrounded by a cloud of electrons. The nucleus is $10^5$ times smaller than the atom, and electrons are much smaller still, so most of the atom is indeed empty space.
Atoms themselves can touch each other, as in molecules or solids/liquids, or they can be separated by empty space as in gases. When solids and liquids are heated, the atoms stay in contact with each other. In gases they can separate further if additional space is provided, for example by enlarging the container the gas is in. Heating the gas in a fixed container will increase the pressure of the gas, as the atoms/molecules are moving faster at a higher temperature.
When atoms touch, it's the electron clouds that do so. These same clouds also generate the binding force that binds atoms into molecules and solids.
Microscopic particles are much smaller than the typical distance between them. However vacuum is typically a macroscopic region with no or few particles. So the answer can both be yes or no. Is this just a wide philosophical question or there's some background?
If by vacuum you mean that there is no medium present than the answer is yes