Getting state names a route goes through

If you are looking for web service that magically solves your problem - I'm afraid I don't know any.

What needs to be done is:

• download the routes from a suitable routing engine, ideally in GeoJSON format (observe potential usage policies - but 100 routes, if queried with a little delay, shouldn't be an issue
• prepare a data set with state boundaries - you can get them from OSM but it's probably easier to find a shape file on a random web site
• have the computer detect which state polygons intersect with the route

One way to automate this is loading your routes and state boundaries into a PostGIS database. Then (let's assume you have loaded the routes with an ID into a "routes" table and the states with their names into a "states" table) the database can do the work for you with a simple query like

SELECT routes.id, states.name
FROM routes, states
WHERE ST_Intersects(routes.geom, states.geom)
ORDER BY routes.id;