It's almost always the DBMS that's the bottleneck when websites drop from traffic. As long as you don't do anything fancier than primary key look-ups you're probably fine though.
I survived handling the search search traffic generated by this thread[1] on PC hardware off a residential broadband connection without any sort of degradation. Only time I've gone offline from traffic was when Elon Musk tweeted a link to one of my blog posts, and that was just a short temporary blip.