I have it supplied with power, it will not process bauxite or lapis, however it would process water.

I am not sure if this is me not understanding how it works or if it's a bug

Mod version is Modern Industrialization 1.6.8 for fabric 1.19.2

It processed one set of lapis dust after I put it down and then just stopped

By default, you need 10 bauxite, 18 lapis or 23 emerald dust in an Electrolyzer's input slot in order for it to process any; you can check the exact numbers for your modpack/server in REI/EMI.

Oh, I didn't know that, my server's REI is broken. Thank you.