What is a Hippogriff?
You appear to be taking the half-eagle/half-horse description literally. In this case, it's an omnivore where the front half and back half have different digestive needs. However, that's really unlikely to be the case. My answer will be from the point of view of horses because once the math's established you can apply it to any dietary requirements you wish. I bring this up because many a good story has fallen upon deaf ears for the lack of originality. Therefore, what, really, is a Hippogriff in your world?
The Dietary Requirements of
There is an issue, but I'll deal with that a bit later.
And before we begin: remember that I'm oversimplifying a remarkably complex process.
Merck has an excellent paper on the subject of horse nutrition that covers everything from water to lactation. There's more to keeping a large animal alive than just the proverbial bread and water. Dietary requirements change for the different phases of the animal's life and (in your case) the different kinds of combat the animals may experience.
A sedentary animal needs 5L/100Kg/day of water. Everything will increase this amount. Merck recommends unlimited access to clean water. That's a bigger issue for your flying island than food because water will be needed for everything: humans, animals, agriculture, driving the engines of manufacturing, sewer/cleaning... just to name a few.
Most of the believability of your story will not revolve around the science of keeping Hippogriffs alive. It will revolve around the believability of your water sources.
According to the USGS an inch of rainfall on one square mile results in 65.78 million litres of water. This sounds like an enormous amount until you realize how much water is needed for farming. An answer to this question suggests 3.5 acre-feet/acre of water per year. 640 acres/square mile: that's 2.76 billion litres annually or 42 inches of rain. (Remember that rainfall requirements depending on soil and crops will vary wildly from this figure.)
This, of course, depends on your soil foundation, top soil quality, what food you're actually growing, where your island is (hot? cold? desert? jungle?), and your altitude (closer to the sun, faster evaporation, though it has little to do with the sun and a lot to do with air pressure). If you need to store the water, you need 2.76 million cubic meters of space. Granted, that sounds bigger than it really is. It's a fraction of the size of a city resevoir (on the order of 1/30th - 1/100th). But, you stil need to make the space, and I don't know how thick your island is. Lakes and ponds might have to be bigger than usual because you don't have the depth to work with.
Merck has tables that indicate your average weight animal needs 16.7 million calories a day if idle and twice that for heavy labor (like combat). That sounds like a lot, but as it turns out, there's a difference between calories, Calories, and megacalories. From the previous link we find:
Once the hay test is completed, the results will show the digestible energy in the hay. It will appear on the test as "DE, equine Mcal/lb." For example, if the test reports a reading of 0.78 DE, equine Mcal/lb it means each pound of hay will offer 0.78 megacalories.
Using the adult 1,100 pound horse that is doing moderate work how much of this hay would need to be consumed? He needs 24.6 Mcals per day. Divide 24.6 by 0.78 which equals 31.5 pounds of hay. This horse is going to be in bad shape - there is no way he can eat that much. Better hay needs to be found (the best thing to do) or a concentrate needs to be fed.
And from this site we discover how much hay/alfalfa/etc we need. For that 1,100 pound horse and assuming we get but one crop per year (REALLY IMPORTANT ISSUE) we get 0.03*1100*365/2000 = 6 tons of feed per animal or 1,200 tons for your cavalry. Reducing this paper from Purdue University to the basic facts (and ignoring the complexities of soil quality, rain, sunlight, etc.) you get 3.4 tons of feed per acre per harvest.
Putting It Together
Assuming one harvest per year, you need 353 acres or 0.55 square miles just for animal feed and the ability to aquire or store 1.52 million cubic meters of water. (It's the same amount of rainfall, though, because of the change in acreage: You're still looking at 42 inches of rain, unless you allow space to lie fallow as a rain shed, which is a good idea. Farm an acre, use 3 for rain shed, now you need only 10-12 inches per year.).
One More Thing
But, there's that issue I mentioned. Hippogriffs fly. That means the high-energy requirements of birds. This is a pull-it-out-of-thin-air guess, but you'll likely need to double the food intake to sustain regular flight. That doubles the requirements of the previous paragraph.
But It's More Than Just Land — it's transport
The second biggest problem you face (next to water) is transport. What happens in a drought, or if some disease hits your crops, or you choose to reduce your landmass so that you're only self-sustaining for short periods of time during war? You need to move mass (water and food) from the surface to your island.
Some gut reactions suggest a horse can pull up to 2X its weight, but really shouldn't carry more than about 15% of its weight (150# for a 1,000# horse) and generally can pull about 1.5X its weight.
The problem is that you need to fly everything to that island. That means we're dealing with "carry" figures, not "pull" figures. Unlike a cart, which distributes the weight on the wheels thereby reducing the actual force load on the horse. Your Hippogriffs must carry the load as if it were on their back.
If we assume you only use your Hippogriffs for transport, and that one rider can wrangle four cargo carriers, then you can move (on average) 120# per Hippogriff or 9.6 tons if the whole fleet is used. You need 125 trips (of the entire fleet) if you need to fully stock for a year. Take you, what, at lest a month... maybe two?
Another chunk of believability in your story will be emergency preparedness, which boils down to transportation. How do you haul cargo to your island? You will need resources for animal maintenance, building maintenance, human maintenance, news & intelligence, and a host of other things. Yes, you can make most of it yourself (assuming nothing goes wrong), but you're just one natural disaster away from everybody dying if you only have 200 fighting birds. You'll need a reasonably comprehensive transport system. If you're only using hippogriffs, you'll need another 100 for regular transport and probably another 100 to swap out animals who are sick, injured, pregnant, tired, etc.
Note that I just doubled your landmass.
Your Last Problems: Metal, Wood, and Coal
Finally, a floating island will have little in the way of mining or logging. You'd need a floating state to have enough forest to be self-sustaining with wood, and even if you created a big enough floating island to have mining, it's a finite resource.
You need to move metal and wood to your island, exacerbating the transport problem because they're both heavy.
On top of that is the need for heat for everything from comfort to cooking to metalworking. I'm assuming coal for convenience.
And that's assuming you're not building your fortress/castle out of stone. Granted, after it's built all you need is maintenance... but rocks are heavy.
Water and transport... they're the real problems.
That was a book. If you read it and all the supporting links, you're well educated! If not, here's the short and skinny:
You need about 4 square miles dedicated to food for your Hippogriffs. One for actual production, three for rain shed and rotation.
You need about 0.33 square miles dedicated to water storage... assuming your island is funnel-shaped and every drop of rain water is channeled to the lake. If not, it'll need to be bigger. And while I'm thinking about it, attacking your island from underneath to punch a hole in the lake would be a great tactic for your enemies....
You need 100–200 Hippogriffs in addition to your cavalry for supporting functions including transportation and down-time replacements.
You need regular transport (as in nearly daily transport) to keep non-renewables in stock including metal, wood, stone, and burnables like coal.