Here is my understanding of the situation.
The official website hrp.org is selling a limited number of advance tickets at a slight discount. These are actual tickets which are mailed to you. So you do not need to queue to buy tickets, although the website warns that at peak times even one with a ticket in hand may have to wait for entry. These advance tickets are indeed sold out for some dates in June.
The visitbritain website is selling prepaid vouchers with an open date. You still have to queue to exchange them for a ticket, so it is not a "skip the line" and not a guaranteed entry. Perhaps there is a dedicated queue for the exchange, apart from the general ticket purchase queue, but I cannot tell from the website. The advantage of the voucher is that you get the advance (discounted) price of £17.10 ($21.50) or less if you qualify for concession price. But you still might face a long line to exchange your voucher.
There may well be a different line or ticket window for voucher exchange, as the official site mentions a collection point for advance tickets which are bought online but collected onsite (presumably because there is not time to send them by post). On the booking page, it says " if you are collecting tickets onsite, you must collect them at the White Court." I did not work through the visitbritain site to see if there is a similar instruction. You might try the "contact us" button on their website to ask about this.
So to answer your question, it appears that at least the visitbritain site is "legit". But you must understand that you are not getting a ticket for entry on a sold-out date. Youbare getting a prepaid voucher and permission to queue. The question is, which queue?
My preference if Inwere doing this would be to choose a date when there are advance tickets available, and buy them far enough ahead there is time for them to be mailed. But if this is not possible, the voucher exchange will still give you the advance price---and the queue.