In short, yes, there is a way. Here's one of them.
Provided the boundaries are all in the OSM dataset, you can obtain the data you're looking for by using Overpass Turbo. Using Overpass does require that you be familiar with the OSM tagging format, but once you build a query, you have the option of exporting the results as a GeoJSON.
In OSM, boundaries are indicated by the key/tag combination
boundary=administrative. This is always paired with a second key/tag
admin_level=*, which indicates the kind of boundary. The values in
admin_level and their administrative equivalents (x = State, y = County, etc.) varies by country.
In the Overpass Turbo interface, you can use the Query Wizard to build your query. For example, here is a query to grab all administrative boundaries in Iceland:
And here is the resulting output:
As you can see, there's a lot going on here. By selecting features, though, I can see how the
admin_level key is being used here. I could either use this output as is and filter it myself, or I have the option of specifying a particular admin_level in the wizard by appending an "and" statement.
Once your results are to your liking, you can select "Export" in the upper-left, where you can then specify the GeoJSON format and download.
If you can't find the results you're looking for, it could be that the data is not in OSM. In that case, you'll have to keep looking for some other dataset that you can convert to GeoJSON. You might try seeing if the country has a GIS portal. Also: consider contributing this missing information to OSM if you find it!