So... I wanted to get into speedrunning TLoZ: BotW on my switch but then I remembered: You need to record them... And post them... And time them... So what gear do you recommend and where could I get the best price?

For a timer, you could use any regular timer on your phone or computer to time the run. For recording, you could use your phone camera to record them as long as the screen is visible enough to distinguish what's happening. To post them, just create a YouTube channel and post the videos there. This literally costs zero dollars so this is probably the easiest and most accessible option.