How can I query polyline data for a road?

It is possible, but only if you set up a suitable server yourself. You will want to import data into a PostGIS database e.g. with imposm or osm2pgsql, and then write some code (e.g. in PHP or another scripting language) that lets the client request something and it replies with a geometry.

The existing services - most notably Overpass API but also the XAPI instances you mentioned - will be able to answer simple queries like "give me all objects have ref=A7 in this bounding box" or "give me all members of a route relation called ref=A7", and this might be sufficient to build a prototype, but you won't be able to build a proper application with that for a number of reasons:

• in some cases you might have to do different searches to find the object you want and it will be too time consuming to run all that from within the browser
• since these services don't reply with a complete geometry but instead with a series of nodes and ways, you'll have to parse and process that in the browser which might be time consuming as well
• especially if you are "zoomed out", these services will return the full, detailed geometry of the object having tens of thousands of nodes and consisting of many parts, when it would be much easier for you to have an assembled and simplified geometry
• if your application has more than the occasional visitor, you might be unduly placing load on the volunteer-operated XAPI/Overpass services.

In addition to what has already been discussed, there's also the routing engine at project-osrm.org which already has some neat features e.g. producing a simplified geometry, but with that engine you can only acces the fastest route between to points and not something arbitrary like "Autobahn x" or so.