Web app architecture for third party API calls

I'm building a simple search engine that takes a user submitted search query as input and outputs a list of appropriate search results. Search queries ultimately get sent to a third party API, which does the heavy lifting of generating search results.

There are 2 ways I can handle this workflow:

1. 
   

   My server takes user requests, queries the third party API, and returns results to the user

   
   

   or

   
   


2. 
   

   Shifting this responsibility to the client-side; the client queries the third party API directly.

   
   

What are some considerations when choosing between these 2 methods?