Although Awaji Island is one of most popular Cycling place but you can not expect same type of Pro Shop like in Osaka or Kobe.
If your Road Bike got Mechanic trouble, there are several spots you can ask to survive.

Cycle Ade points in North to center of Awaji island

Bikers Cafe circolo:
You can expect almost all mechanic trouble.
Even Di2 Battery Charge, minor 26inch Clincher Tire, or Tubular tire, you can get here.
In case serious trouble, suggest to take their Rental Road Bike service. TEL:0799-64-7533. Pick Up service can be available, along to the situation (Busy or not so) in Cafe Time.

Bikers Cafe Circolo

Aeon Bike:
Cycle Corner in Big Super Market.
Aeon Bike in Googlemap
However, you can get most conventional parts here, like Tube/Wire/Tire or even, SPD Cleat.
To Fix your bike, better to consider to do it yourself. 

You can get spare parts at AEON Bikes

Cycle Ade points in South Awaji island

Awaji Kaijyo Hotel:
Very Japanese Classic Style Hotel preparing Bicycle space, display spare tube and Air pump. They also provide spacial hotel rate designed for cyclists. You can stay at their annex with quite cheap price in weekday.
Suggest to contact through mail since they have no English speak service at regular bases.

Bike rack is preparing in front of Annex

Giro de Awaji:
Situating at South Awaji, Fukura town (Googlemap).
Most conventional Road Bike parts, such as 700C Tire/Wire/Tube can be available. In case of Mechanic Trouble, you can ask the Cafe Owner to repair your Bike (If his Cafe is not so Busy). In case of Serious Trouble, you can ask Rental Bike. Even the shop closed, you can purchase above bike parts with their vending machine!! 

Cycle parts vending machine…

Cycle Ade points in West coast road in Awaji island

Cycle Shop SAITOU:

Also, Most conventional Road Bike Parts can be available at here.
Also, you can ask pick up service if you are NOT so far away.
English cannot be expected.

Find out someone to explain in Japanese…. Address: 181 Kozumichi, Matsuho, Minamiawaji TEL:0799-36-2124 (Google Map)

It is barber Shop. But you can expect his Good skill if the Barber is not so busy… 700C Tire/Tube and wire can be available.
But again, this is Barber…. Just for your emergency case…

Also No English speaker available.  
104-6 Kanda, Asano, Awaji City TEL 0799-82-0809  (Googlemap)

Ride safe in Awaji Island and enjoy even your mechanic trouble.
The trouble may be the chance to touch with Local heart!!!