Excellent advice above.
I'll add one more thing you should consider -- though it's merely a potential explanation for what you are seeing, rather than a solution to that...
It's still early. It's just the middle of January. The places you are looking at are almost all small, low-key, family-run operations, not the Hilton. They are in a sleepy, fairly obscure place -- sleepy and obscure by the standards of the tourism industry, which is a large part of their appeal. Inis Mor is sleepy even in high season, in January it barely has a pulse. But that sleepy-off-the-beaten-path quality also requires that we adjust our expectations a bit. Right now is the depth of their "low season." Some of the folks running these places may be away on holiday, or just not actively tending to their business every day. Some probably not have loaded their availability into web-based booking engines yet, because they're not in the mood to, or not ready to, or just don't feel like they need to. They'll get around to that later.
It's true that accommodations are limited there, but just because you are not seeing availability in June does not necessarily mean that every place is completely booked up already. Look for email contacts for the places you see listed with no availability. Email them and ask if they are booked up for your dates, you may find that they just have not started accepting bookings yet.
The travel industry is booming, it seems everyone is going places and demand is sky-high, so those of us who prefer to nail down our accommodations early (I'm definitely one of those) have a natural tendency to book everything super-early (to beat the competition for the limited places). But not everyone in the worldwide tourist industry has adopted that cut-throat, competitive approach. Some folks are still taking things easy, because they are confident that once summer arrives, the people will come.
Don't give up, try and ping places via email and see what that yields.
(Afterthoughts: Some places might have a two- or three-night minimum in high season...try different dates and see if anything magically opens up for 2 or 3 nights, during different weeks; also, don't know how many are in your party but if you are looking for more than 2 people that is always harder, so try looking for just 2 and see if anything appears open....if you try for just 2 people for 2-3 nights and you see absolutely nothing available all summer, that would be a strong indication to me that they are just not taking bookings at all yet...a hopeful sign).