I will answer the question by comparing empty space and
vacuum with the properties of the interior of an atom.
I will write at the level of high school physics in order to make my answer accessible.
My conclusion will be that the answer is "no": it is not true that the interior of an atom
is mostly vacuum or empty space, and furthermore
such an idea conveys a thoroughly misleading picture
of the nature of atoms. The main point is that the electrons
are smoothly spread throughout the interior of each atom, and they
carry enough mass and charge to make it misleading to compare
the situation to empty space.
First let's see what we mean by empty space. We mean of course "nothing is there". To flesh this out a little, consider the case of an ordinary
gas at standard temperature and pressure (STP). This is not a vacuum, clearly,
since the pressure is quite high. We might speak of "vacuum" once the
pressure is below a millibar (100 Pa); at STP the pressure is about a bar ($10^5$ Pa). On the other hand it is true to say that such a gas is "mostly empty space" in that the mass is concentrated in the molecules, with almost no mass in between the molecules, and the volume occupied by the molecules is a small fraction (about a thousandth)
of the total. On the other hand, if you place a mass-detector anywhere in the gas, then it will register some mass very quickly, because the
molecules will soon hit it. So to say that a gas is "mostly empty space" is helpful to get the intuition that the molecules can move
freely with a large mean free path, but it can mislead in some respects.
The density of an ordinary gas at STP is about 1 kg/m$^3$.
Now let's think about the interior of an atom. I have in mind some ordinary atom such as carbon, and ordinary locations inside the atom, so not at the nucleus and not far away; one could pick a location about half a Bohr radius from the centre, for example. Let's calculate
some properties.
First, the mass density. This is the mass density owing to the electrons which are present. Their mass is spread throughout the atom
via their extended wavefunctions, and the average mass density can
be estimated by noting that an electron is about 2000 times lighter
than a proton, and a typical atomic nucleus has as many neutrons as protons, so the electrons contribute about one part in 4000 of the total mass. The density of a solid element such as carbon is about 2000 kg/m$^3$
so we can estimate that the density owing to the electrons at a typical place in an atom is about $0.5\, {\rm kg}/{\rm m}^3$. An estimate using the atomic radius of carbon gives the value
8 kg/m$^3$, suggesting that our previous value was an underestimate
because there is some space between the atoms. Anyway the main conclusion
here is that mass density
at a typical spot inside an atom is similar to the average mass density of a gas at STP.
To get the charge density, we multiply the mass density by $q/m_e$,
the charge to mass ratio of an electron,
which gives about $10^{12}$ coulombs per cubic metre.
This is a huge charge density by everyday standards. (For comparison,
a typical
1 microfarad capacitor charged to 1 volt carries a micro-Coulomb in a volume of order $10^{-7}$ m$^3$, giving a charge density $10$ C/m$^3$.)
Next let's consider the flux of matter---the rate at which mass
will approach and hit a detector if we were able to place a mass-detector
inside our atom. The electrons have speeds of order a few times
$\alpha c$ where $\alpha \simeq 1/137$ is the fine structure constant
and $c$ is the speed of light. The flux (mass crossing unit area per
unit time) is therefore around $8 \times 3\times 10^8 / 137 \simeq
10^7$ kg per second per square metre. Needless to say, this is a huge
value in everyday terms.
Next let's enquire into whether or not there is "empty space" in the sense that there is room to put stuff inside an atom. The original statement perhaps comes from a desire to compare an atom to a gas,
using some notion that electrons are point-like in some sense, with
room in between them.
To address this question we need some more advanced physical ideas, to
do with the Pauli exclusion principle. This is an important result in quantum physics, which says that particles such as electrons cannot overlap one another completely. To be precise, in any given spatial situation there can be at most two electrons having that particular combination of position and momentum.
What this means in practice is that there is no more room for low-energy
electrons in any atom. If the atom is a hotel, then all the rooms on the
lower floors are occupied---completely full up. Thus the
space inside an atom is completely unavailable to further electrons
unless they move quickly. This is about as far from "empty space" as you can get. It is "complete and utterly full-up space", as far
as low-energy electrons are concerned. But this does not exclude electrons altogether, as I already said. If they are moving quickly
then there is room for further electrons to get into an atom. They
won't stay there---they would have to form a beam passing through;
they are visitors to the guests staying in the hotel. For an extra electron
bound to an atom (making a negatively charged ion), the wavefunction of
the extra electron does get inside the atom a bit (it penetrates the
core, as we say), and this can be compared with a visitor rapidly
visiting again and again.
What about other types of particle---say, neutrons? They can more
easily enter an atom. But is the experience of a neutron sitting inside
an atom anything like the experience of a neutron sitting in empty
space? Hardly. They would be continually bombarded by that high flux
of electrons we calculated just now, and they would notice because
although they carry no electric charge, neutrons carry a substantial
magnetism, and this leads to an electromagnetic interaction between the
neutron and all the nearby electrons.
Now let's summarize.
Electrons in atoms behave in ways that classical physics cannot
account for; we require quantum physics. As a result, when we talk about
atoms in everyday language, we are trying to convey in everyday terms what quantum physics says is going on. Among the things that quantum physics tells us about the interior of an atom is that the electrons are
smoothly spread out, such that the probability of an electron being present at any given moment is non-zero throughout the interior of
an atom. We can flesh this out a little by calculating properties such
as mass density and charge density and flux. The mass density of
the electron cloud of a typical atom is similar to the average
mass density of an ordinary gas at standard temperature and pressure.
The flux is huge and the charge density is enormous. Also, it is strictly impossible to introduce further slow-moving electrons into the inside
of an atom, but it is possible for fast-moving electrons to pass through. Neutrons can also enter atoms, and when inside they interact
with the electrons which are there.
In view of the above, it seems to me that it is misleading to say that
the interior of atoms is anything like either vacuum or empty space.
It really isn't. But it seems that this idea has become lodged into popular
presentations of science. It will take some effort to dislodge it.
I now wonder where this idea came from in the first place.
I think possibly it might have originated in the early attempts to
model atoms via classical point-like particles, or perhaps it is
descended from the "fly in a cathedral" image, which is a correct
statement about the relative sizes of the atomic nucleus and the
whole atom. The "fly in a cathedral" seems to imply that the rest
of the "cathedral" is empty, but it is not. It is full of electrons.
