I have a last generation 105 5800 groupset currently set up with mechanical disc brakes. I'm thinking about upgrading to hydraulic brakes. Do I need to change my shifters in order to do this?
As an alternative to the expense of upgrading to hydraulic STIs, there are cable-to-hydraulic adapters available. I recall seeing some Genesis road bikes equipped with them in a local bike shop.
We don't do product recommendations on Stack Exchange, but here's some examples of what I'm talking about.
I have no personal experience of such adapters, though, so you'd need to do your own research!
STIs include the internals actuating the brakes, i.e. the mechanism pulling the cable or the reservoir including the brake fluid etc. For this reason, they are not interchangable, even within the same generation of a groupset (if both mechanical and hydraulic options exist, for example Ultegra R8000).
You need ST-R7020 levers including matching BR-R7070 calipers, the rest of your groupset (R5800 front and rear 11-speed derailleurs) should still be compatible.