I'm assuming here that you're dealing with a company that's known to, presumably because they took your wedding photographs - or those of a friend or relative - at a ceremony in Cyprus.
You shouldn't need to give your sort code. However, just as with an debit card payment in the UK, the studio will need the long card number, the expiry date (and possibly the start date) and the three digit security code on the back.
Sending such details by email can be risky, as email messages can be intercepted by fraudsters. (If you must do it, put the information in a Word document and send that as an attachment. Then automated software won't spot that there are bank details in the message, in the same way that BT's software did).
The easiest way, by far, to get round the problem is simply to call them and give them the details over the phone. (As long as you know it's a reputable company, there's no greater risk doing that for a purchase from Cyprus than for one from a UK company).