Solution

My approach

Initially started by making a request to Foursquare endpoint in order to see the shape of the data returned and create a SearchResultType from this.
Build a basic structure to the app... a form and some basically markedup search results.
With the basic data fetching, I moved quickly onto working with localStorage to persist results from queries made with the form.
Finally I went back to tidying up the look and feel of the application, adding in a drawer model to display "more" information about a query entry.

Given more time:

I'd write some tests. previous roles did require me to write tests with react-testing-library so I do have some experience with this.
I'd think about whether pagination could be required to render larger amounts of data.
I'd address more of the a11y where required too given more time.
Break my working into regular commits, I was far too conscious of spending too much time on the project.
Probably shy away from CSS modules in future, they felt restricting in the way I should structure things. This might also be a symptom of having never used them before.
I'm a big proponent of utility classes to build applications at scale. Something like Tailwind or similar could have been used here to save a bunch of time, but I wanted to address the requirement "Showcase your CSS chops. Try and do some amount of original CSS styling".
I'd also address the fact that the localStorage hook doesn't handle deleting entries at the moment.