Short Answer yes, there may be side effects.
Long answer. They are looking at more than just sulfates. Initial research has shown an optimistic lack of side effects for sulfates but we won't know until we actually test it and testing over a 1km area, something attempted multiple times, has been blocked for environmental concerns. Concerns exist because we can't test it because it's blocked for environmental concerns.
The thing to keep in mind is that the original idea was
- proposed to shock the scientific community into researching multiple avenues by Lowell Wood in 1999 (I could not find the source but it was a Wikipedia article so take it as you will.)
- is based off the cooling effects of volcanoes.
The second point is the most important because we have a natural phenomenon to work off of. there was a volcanic eruption in the 90s that lowered global temperatures by ~1.5 degrees (I believe it was C) over 18 months before things returned to normal. This means we can study the effects of volcanoes and predict the effects of the aerosol injection. Side effects of volcanoes include large impacts on vegetation and cooling but little else is cited in the articles I've seen. Fortunately, they believe that aerosolized injection into the upper atmosphere will avoid affecting plants very much if at all as the particles will mostly stay up there. However, there is concern that it might modify our atmosphere in some unknown way.
I have seen some articles saying sulfur emissions in China/asia might actually already be helping lower global temperatures a little but the idea was theoretical and the articles mostly seemed like clickbait.
Really we aren't sure and need to do some testing to find out. However, this will be small scale testing that will ramp up slowly.