You're going to be out of luck. If you go to your YouTube account settings, and go to the Privacy section, you'll see this option:

Keep all my subscriptions private

If somebody has chosen this option, they can subscribe to you and they will not appear on your subscriber list. They're subscribed; they'll get your videos in their feed; your subscriber count will go up by one; but you will not be able to see that they are subscribed to you.

I just checked the subscriber list of a very small YouTube channel that I have access to. The channel only has 44 subscribers. The list of subscribers I'm able to see only had 27 people, which implies 17 of them had this option enabled. That's 39% of them. Obviously this is a VERY small sample, but this isn't a channel that is especially technology-oriented or privacy-oriented, so I don't have any reason to think these numbers would be atypical.

I don't think you thought through the terms and conditions for your competition, and I think that you're probably best off replacing your original entry requirements.