I've relocated to an apartment with considerable mold in a couple of closets and bathroom.
Is there a way to remove this mold with a biodegradable solution?
There are plenty of biodegradable anti-fungals.
For example, cetrimide is water-soluble, anti-fungal, anti-bacterial and biodegradable. White vinegar also has anti-fungal and anti-bacterial properties
But that's only part of the solution.
The other part is to stop the mould returning.
To do that, you need to increase the surface temperature of the areas that are getting mouldy, decrease the humidity (or do both). The aim is to get the relative humidity of the air at these surfaces down below 80%, and ideally to 50%.
That will be some combination of: internal wall insulation; external wall insulation; heating; dehumidifier; better ventilation.
Try scrubbing with vinegar. You can try adding baking soda to the vinegar as well.
I've used vinegar with great success to get mold out of clothes by soaking them in a 4:1 water:vinegar mix overnight.
As preventative measures in addition to reducing humidity, increasing airflow and sunlight exposure, I also find it useful to use oil warmers, diffusing essential oils, which have anti-fungal or anti-bacterial properties (e.g. lavender). The increase in humidity (the lamp uses water to diffuse the scent) is negligible compared to the effect.
I am not a big fan of incense, but that would be another alternative, I guess.
I notice that vinegar has been recommended in two other answers. While it is true that acids and the low pH of vinegar are helpful agains mold it should not be applied to all surfaces. Used on various plasterings (typically of a high pH value) the vinegars acid is neutralized and fails in its anti-fungal properties.
Isopropyl alcohol is recommended to be used on such surfaces.