There isn't really a database as you request. Finding the ages of stars is difficult. Only one star has an accurately known age - the Sun. That comes from radioisotope dating of meteorites. For other stars we must rely on models to a greater or lesser extent and we can only estimate an age if the star has a mass or is in a phase of its evolution where things are changing rapidly enough to give some handle on how old it is. For stars at the mass of the Sun or a little bit bigger, one can use evolution in the Hertzsprung-Russell diagram. Stars become more luminous as they burn through their core hydrogen and precise measurements can give an age to about $\pm 1$ Gyr. These are the data that Snaith et al. (2015) use in their paper.
So what Snaith et al. do is they attempt to constrain the star formation history of our Galaxy (or at least stars in the disk of our Galaxy) by making a Galactic Chemical Evolution model that predicts how the abundances of silicon and iron change with time and compare the results of their model with the observed silicon and iron abundances in this set of solar-type stars with reasonably well-estimated ages. Iron and silicon are used because they provide differing constraints. Iron is produced during the evolution of relatively low-mass ($1.5 - 4 M_{\odot}$) stars that live their long(ish) lives (1-10 Gyr), become white dwarfs and then some of the white dwarfs explode as type Ia supernovae. Silicon is produced by massive stars with short lives ($<0.1$ Gyr) and spread promptly into the interstellar medium by core-collape type II supernovae.
A representative lot from Snaith's paper is shown below. The star forming rate is shown in plot (a) and the match of the model to the observational data (they are trying to match the solid green points here) on chemical abundance is shown in the other plots.
They test a number of models and different assumptions, but it appears to be quite a robust result (found in other studies also) that there was a burst of star formation 10-12 billion years ago, a lull at 8-9 billion years ago and then a more constant, lower rate over the last 7 billion years. The very low rate in their model over the last 2 billion years is unlikely to be true and is probably an artefact of the age uncertainties in their data points (e.g. stars with age of $1 \pm 1$ billion years) and we certainly see plenty of star formation in the disk of the Galaxy today.
Contrary to what you say in your question this does (almost) represent the age distribution of stars in the Galaxy. That is because the vast majority of stars (90+%) are of a solar mass or lower. Such stars have main sequence life times of 11+ billion years, such that almost every one of them that was born is still a main sequence star now.
There will be a small reduction in the numbers of the oldest stars, because a small fraction of those ($<10$%) will have lived and died and also because a small fraction of the oldest stars may have escaped the Galaxy entirely (due to various kinematic processes that cause their orbital speeds to change). There is also an effect whereby if you take a volume close to the Sun, the oldest stars will be under-represented because they are distributed more broadly around the disc midplane for the same reason.
But by an large, the upper left plot in the picture is roughly what we expect the age distribution of stars to be in the Galactic disk.
There are a small population of very old stars ($12+$ bilion years), perhaps 1%, that are distributed more widely in the Galactic halo.
