I did drug discovery in grad school -- finding leads is generally not something suitable for small labs, but studying them is. A big source of compounds that actually work isn't high-throughput screening, it's isolating the active ingredient of plants and fungal matter that has the desired properties in nature, which is something that can in theory be done in a home lab (cultivating eukaryotic cells is difficult compared to bacteria but there are protocols and cell lines that aren't too expensive). This was how pretty much all science was done before molecular biology was a thing, and it gave us very useful compounds, including penicillin. (For the sake of the poor critters, please don't do animal research if you have never worked with lab animals before).

Most of figuring out which drugs are promising is biochemistry (and clinical trials) not so much organic chemistry. I wouldn't say organic chemistry is super inaccessible (given people with 0 education seem to be able to pop up kitchen counter meth labs in no time), but doing it with a purpose that isn't "follow this recipe" is difficult because a lot of stuff that looks like it should be biologically active doesn't actually make it into the cell in the first place. But I may be biased and I'm sure there are organic chemists around that have an idea of how amateurs could go around doing some small science in that area.