Do planets orbiting stars emit gravitational waves?
Yes, but undetectably. The Earth-Sun system radiates a continuous power average of about 200 watts as gravitational radiation. As Wikipedia explains, “At this rate, it would take the Earth approximately $1\times 10^{13}$ times more than the current age of the Universe to spiral onto the Sun.”
The Hulse-Taylor binary (two neutron stars, one a pulsar) was the first system in which the gravitational decay rate was measurable. It radiates $7.35\times 10^{24}$ watts as gravitational radiation, about 1.9% of the power radiated as light by the Sun.
Yes, two bodies orbiting each other like this will indeed emit gravitational waves, regardless of whether or not they're compact objects like neutron stars or black holes. Obviously, most exoplanets will not emit strongly; a planet-star system generally involves large separations and non-relativistic speeds. Therefore, as G. Smith noted, while all such systems emit gravitational waves, the radiation is largely insignificant.
It's been proposed (Cunha et al. 2018) that some exoplanets with extremely small semi-major axes ($a\sim0.01$ AU) could be sources of gravitational waves that would detectable in the near future. As in most of these cases $a$ is large compared to the sources LIGO has observed so far (compact objects in the process of merging), these waves would be relatively low-frequency ($f\sim10^{-4}$ Hz) and would fall in the regime of long-baseline space-based interferometers like LISA, not ground-based interferometers like LIGO. Some exoplanets could reach peak strains of $h\sim10^{-22}$, which is indeed above LISA's sensitivity curve at those frequencies. (Compare this to the binary systems LIGO has observed so far, with $f\sim10^2\mathrm{-}10^3$ and $h\sim10^{-22}\mathrm{-}10^{-21}$ at peak.)
The authors note that in these systems, orbital decay is indeed occurring, but at lower rates than, say, famous orbiting compact objects like the Hulse-Taylor binary pulsar. Over long timescales, this decay should be detectable. In a few systems, the period decay are comparable to the Hulse-Taylor binary, within a factor of a few, although the gravitational wave luminosities remain lower by a couple orders of magnitude or more.
G.Smith and HDE 226868 gave good answers.
I would add that, in the Solar system case, the gravitational waves are clearly not the dominant factor in changing the (Keplerian parameters of) orbits. Momentum exchange between planets, solar radiation pressure, solar wind effects, tidal effects - every one of these (and probably more that I cannot recall right now) are orders of magnitude stronger than the orbit decay because of gravitational waves radiation.