I think you'll need the Ordnance Survey CodePoint data to map your postcodes to a longitude and latitude. Codepoint is available as a commercial product, or there's also Codepoint Open
Once you have the latitude and longitude, then you can use standard algorithms such as Haversine or Vincenty to calculate distances between the points
EDIT
Structure of the CodePoint Open CSV files can be found here. Note that location is held as Northings and Esatings rather than Longitude and Latitude, so it will need converting. There's a number of articles about this on the web, e.g.
http://www.deepbluesky.com/blog/-/converting-os-coodinates-into-longitude-latitude_7/
http://mediakey.dk/~cc/convert-northing-and-easting-utm-to-longitude-and-latitude/
but you need to be aware that OS Northings/Eastings are based on the Airy 1830 ellipsoid rather than the WGS84 model used by Google maps (and most GSM systems). Failing to allow for this difference can put you out by anything between 70-120m between Cornwall and East Anglia.
You can also find PHP functions to do this conversion at the Movable Type site (essential reading for any PHP developer, working with GeoData. I'd recommend adding a couple off columns to your stockist database for longitude and latitude, and a one-off script to update all existing data using the Codepoint data, then modify your insert/update routines for stockists to keep this information up-to-date.
Using PHP, another solution for this conversion is PHPCoord by Jonathan Stott
For higher performance in your database query, do a lookup of lat/long against a "bounding box" before calculating distances. I've explained how to do this in response to a similar query.