Atoms or molecules with spin 1 in the ground state?
Atoms

Atomic carbon with its $\mathrm{1s^2 2s^2 2p^2}$ configuration has a triplet ground state ($S = 1$), precisely because of Hund's first rule.

However, in the context of the Stern–Gerlach experiment, you might run into a problem with orbital angular momentum, as carbon's ground state also has nonzero orbital angular momentum ($^3\mathrm{P}$ ground state, $L = 1$). The behaviour in a magnetic field will be rather more complex and you probably need to take into account spin-orbit coupling.

At the moment I can't think of any atoms with a ground state term symbol of $\mathrm{^3S}$. I actually suspect that it's impossible, but I'm not really up to proving it right now.



Molecules

Triplet dioxygen (as Zhe mentioned) has no orbital angular momentum ($^3\Sigma_\mathrm{g}^-$ ground state, $\Lambda = 0$), but I'm not sure if the inhomogeneity of the electron density would have any impact. (as in, I'm genuinely not sure.)



Ions

The $\ce{^2H}$ nucleus (which us chemists usually refer to as $\ce{D+}$) is lightest stable nucleus that has a spin of 1.

The only issue is that you then have a moving charge in a magnetic field. If you could ignore it, or take it out of the equation, somehow...