You have to contact Buckingham Palace first and it's best to send a photocopy of the marriage certificate to prove the occasion is authentic. The Palace now receives large numbers of such requests so you will preferably need to give them a couple of months' notice. The tribute comes by way of a card, rather than a telegram and if my memory serves me correctly when my parents' received their card it wasn't actually signed by the Queen herself.
Full information is available here: http://www.royal.gov.uk/output/Page615.asp
(The link to the application details doesn't exactly leap out at you. Look above the Queen's portrait).
Yup - that's the link. I ordered one for my parents last May, sending in the relevant details and a copy of their marriage certificate. It came on cue and was much appreciated.
I also found this site for an unusual and individual Greetings card - you might find it useful: