It might be possible.
We've known for around a century (since at least 1933) that ultraviolet light can inhibit photosynthesis and possibly damage photosynthetic mechanisms inside an organism. Phytoplankton in particular have been recent targets of this sort of research, and it's been shown that their photosynthetic systems are negatively impacted by ultraviolet photons - even more so in the case of shorter wavelengths of light (think $\sim$300 nm). Presumably, with a strong, continuous source of ultraviolet radiation, typical photosynthetic organisms like the ones we have on Earth would be quickly killed off.
Now, it's possible that in your world, organisms could adapt to solve the problem of ultraviolet light. As one of those two phytoplankton studies found,
As the cells were grown during long-term exposures to solar radiation, the ratios of repair to UV-related damage increased, indicating their acclimation to UV.
Therefore, if the organisms are exposed to low levels of ultraviolet radiation for millions of years, they will presumably develop a significant degree of tolerance. A way you could make this even simpler would be to replace the Sun with a star that's slightly hotter - say, maybe 7,000 or 8,000 Kelvin, which you might see on an F-type star. That star would emit more ultraviolet light (doing about 2.5 to 7.1 times as much damage to DNA as the Sun does), and so any organisms developing on a planet around it would be more likely to adapt to the high-energy photons, even if the aurora is not the main source of light for photosynthesis. Of course, there's still going to be DNA damage, regardless - a strike against using ultraviolet light from an aurora at all.
Let's talk about auroral brightness. The best source I was able to find was the Australian Government Bureau of Meteorology. They note that in some very rare cases, auroras have emitted fluxes of 1000 kilorayleigh (a unit of photon flux); they describe an aurora like this as
Bright as the full moon. Casts shadows. Very rare.
Of course, the Moon is about 400,000 times dimmer than the Sun, so you would presumably need continuous, intense stellar activity to create an aurora that could come within even an order of magnitude of the Sun's apparent brightness.
Could something like that happen? Perhaps. Some Sun-like stars are known to emit superflares, which release large amounts of energy on timescales of minutes to days. They've been observed in cool F stars, like Omicron Aquilae and 5 Serpentis. It's not out of the question that you could have a hot superflare star that would emit enough energy and ultraviolet light to create the scenario you're looking for. The one problem I can find with superflares is that they only tend to happen once every thousand years or so; perhaps your star could get around this by emitting low-energy superflares more frequently.