Both work. If you want to have the going on emphasis in Dutch, you could add nu or momenteel: Er wordt nu gezocht naar mijn schoenen. …but then translating it back to English I would add now or at the moment as well. There isn't really a perfect 1:1 translation here I guess.
"There is a search" = "er is een zoektocht". In fact that is the same of course. But there is a slight difference between: "er wordt gezocht" and "er is een zoektocht gaande". This flexibility of Duolingo is surprising.