Despite much speculation over time - and especially during recent years - no member of the Royal Family has any official political view. We can probably assume that none of them are fans of any Republican cause or party, but they're not allowed to discuss such matters in public. Not only do they refrain from talking about it, it would be considered unconstitutional for them even to vote.
This has proven difficult for even the most experienced royals on occasion. In the days leading up to the Scottish independence referendum in 2014, the Queen made a passing comment to a well-wisher that was construed as being in favor of Scotland remaining in the UK. Prince Charles, with his lifelong interest in environmental causes, has also veered very close to the line with his statements on climate policy in the past.