A very brief Google search gets you the number 1,192,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (approximately $10^{57}$ atoms)- but in fact this is wrong. That value is derived from the mass of the objects of the solar system (mostly the Sun) divided by the mass of a proton (which is what most of the Sun is made of).

But the question asks for the number of atoms in the solar system. Since the Sun is a plasma ball, most of its mass is not in the form of atoms. We are then left with estimating the number of atoms in the rest of the solar system, for which we need the composition of the various planets. The largest objects in the solar system (after the Sun) are:

Name     radius    mass 
          (km)    (kg)
Jupiter  71492  1.90E+27
Saturn   60268  5.68E+26
Neptune  24766  1.02E+26
Uranus   25559  8.68E+25
Earth     6378  5.97E+24

Of these, Jupiter is about 71% of the total mass of non-Sun objects; everything including Earth is 99.8% of the mass.

To get to the number of atoms, we need a reasonable estimate of the atomic composition. According to this link 80-some percent of the mass of the giant gas planets is Hydrogen and Helium. This means that we would get an upper limit on the number of atoms if we just took the mass of these five, and pretended it was all hydrogen:

Total mass: 2.67E27 kg; mass of proton 1.67E-27; number of atoms on the order of $10^{54}$.

That is three orders of magnitude smaller than the answer you get when you include the Sun. Of course not all of the Sun is fully ionized - see this paper for a painfully detailed calculation of the degrees of ionization. But the photosphere (where the non-plasma parts of the Sun are) is a very small (and low density) fraction of the Sun; including it properly is well outside of the scope of your question, I'm sure.