Is the atom the smallest particle, which takes part in chemical reactions?
Protons definitely participate in chemical reactions.  Free protons are generally not present in liquid water because a free proton is extremely reactive, but in the upper atmosphere or in other situations where the density of matter is low there can be free protons which participate in chemical reactions.
However, I completely agree with Jori that an electron, which is much smaller than a proton, takes part in chemical reactions.  Gain or loss of one tiny electron can completely change the properties of a large metalloenzyme.
How small is an electron vs. a proton? In terms of mass, the electron is 1836 times lighter, but what about in terms of radius. According to Hans Dehelt's Nobel Lecture:

Proton radius [is] $0.86 \times 10^{-15}$m...
Today everybody “knows” the electron is an
indivisible atomon, a Dirac point particle with
radius R = 0....
But is it?

And goes on to explain that it is still possible the electron could have a radius on the order of $10^{-20}$ m.
See also Limit on Electron Substructure
Also, photons, which have no mass, participate in chemical reactions.