So I found the problem. There is a JS script ( probably loaded by your theme ) on your site that also tries to style the dropdowns.
I suspect it’s this script https://github.com/select2, but it’s probably included in a JS file instead of being loaded separately. I don’t see it anywhere in the HTML output.
The easiest way to find out where it comes from is by searching through your theme files for either ‘select2-container’ or ‘select2-offscreen’ both are CSS classes placed on the dropdown from the store locator, but they don’t belong there.
How it works is that the wpsl-gmap.js looks for a div with the ‘wpsl-dropdown’ class inside the div wrapped around the dropdown. Usually there is only one, but now because of the other JS script there are 2 which breaks the category filter.