There's nothing to prevent you asking anyone for any information. However, in many cases, the Data Protection Act will prevent them from actually giving you that information.
It occurs to me though that your tenants will almost certainly be seeking to get credit from people at their new address (because I'm assuming that they're the type of people who will happily borrow money irrespective of whether they can actually pay it back). That means that they'll need to get their names on the electoral roll. Once they've done that you'll be able to visit your local council office and ask to view the electoral roll for the relevant area. (It's a public document. The council MUST let you see it). The electoral roll is listed alphabetically by street names (rather than by the names of voters), so you'll need to look through all of the addresses on the relevant road.