In examples like yours, the word "indulge" requires the addition of the preposition "in". It's because in English 'we usually indulge in something'. There are two ways of complying with this requirement:
(1) "People who are adventurous enough to leave the USA and explore other parts of the world are bound to find breathtaking activities to indulge in".
(2) "People who are adventurous enough to leave the USA and explore other parts of the world are bound to find breathtaking activities in which to indulge".
In (1) we have simply added the preposition "to" after "indulge". Prepositions normally require complements to complete their meaning, and in this case the complement is "breathtaking activities". So think of it as meaning "to indulge in (breathtaking activities)".
In (2) we have added the preposition "in" + "which" to form a relative clause: "in which to indulge", where "which" refers to "breathtaking activities".
Both alternants are acceptable, though (2) is slightly more formal.