If you pay via a site which has the little padlock showing on the toolbar at bottom left, this is as safe as it can be. I use internet payment by credit or debit cards very often and (touch wood) have had no problems. You then can print a proof of your payment too, which you can't do if you pay over the phone.
You might find delays if you send a cheque - you have to mail it, they bank it, it takes several days to clear, usually only then can any transaction be confirmed.
Only the campside can tell you if they will advise in advance if they have availability - they might be able to hold a provisional place until payment arrives.