Can I somehow make my inactive alchemist do alchemy by themselves and advance in their careers by selling the potions as well? I don't see an alchemy skill in in SP skills drop down, so I don't even know if it can be done.

I'm a bit confused by the question and not sure I can be of much help here, but what I do know is you can manually assign careers and its level to inactives, as well as skill levels, through MasterController instead. While SP also has plenty of options, I didn't see any Push option relating specifically to Alchemy and creating potions, as opposed to some others like Buskers, Inventors and Writers.

My best suggestion would be to check that inactive's LTW and traits, and if applicable, choose new ones that relate to Alchemy. This will help in pushing them to do activities related to their job on their own, such as using the Alchemist Station. For example, if you made an Inactive a self-employed Alchemist through MC but their LTW is Superstar Actor, as long as there's the appropriate rabbithole, SP will probably have them switching jobs (if Auto Find Job is set to the default, which is True - at least as far as I know).
Next would be to check if there's an Alchemy Station in their house, so that they'll automatically look to increase their skill and make potions even in their free time, and because it makes it easier for them to be prompted to do so.
Whether it matters that they have the necessary ingredients for a certain potion in order to make it (since they're inactive) and if they'll look to get them, or if they'll look for new recipes by themselves, I honestly can't say. I've made an inactive an Alchemist myself but haven't been keeping up with their progress.

I don't know how to push an inactive to sell their potions at a consignment store. In SP there is, for instance, the Push Photography option that should help push inactives to take and sell photos, which would count for their self-employed Photographer career, as well as others, but Alchemy seems to be missing, might never have been added specifically? Whether or not this means SP can progress that self-employment career like others, I also don't know.
I haven't looked but there might be something you can get to and alter with Retuner? If there's anyone who knows that mod much better and can offer some advice on that. This is as far as my personal knowledge of the game and mods goes. Hope it helps you a little.

Currently the Alchemy career isn't pushed/supported by SP and I don't know of anyway to do that.

Too bad it isn't supported, Thanks for the replies!