I think what's happening here is, you're putting organic search and local search in the same bag, but they're not the same thing.
When someone searches for "[product] near me" in a search engine, local SEO comes into play - you've probably heard of Google Business Profile. Google looks for businesses in the database it has for a certain geolocation. It gets this geo from the user's browser/device location, meaning these features must be on and accessible to the search engine.
You have this feature on your webpage, it seems, for places that sell your product. But Google won't be using your on-page search feature; it has its own. So this feature is still useful, but for your customers and website visitors.
For search engines, you'll have to do at least two things:
- Make sure that your website is localized with proper hreflang tags.
- Include a page with a list of businesses that sell your product in different regions. Break it down by region, using h-level headings as appropriate, and list all the stores and their locations. Make sure your page can be read by all popular search engines. Keep it updated. Here's an example of what I mean, except this is a brand listing its own stores, but you'd list the stores that carry your product.
There may be more things available to you here, such as being included in certain business directories, but that depends on your product and marketing strategy.