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That thermostat model number does not appear to be the communicating one for that system. It would be either an RHCTST-500 (501 or 551). The 0 simply means it is not receiving a call. If we are just missing the thermostat number by some reason, and it is in fact a communicating thermostat then it would be losing communication. Try to post the whole thermostat model number so I can tell if it's set up communicating. What is the indoor unit (air handler or furnace) model number?
It would appear there is a loose connection in the thermostat wires at the outdoor unit in all likelihood. You should see a bunch of blue or orange wirenuts under the control box in the outdoor unit. Let's start there by taking them apart and putting them back together 1 by 1. Power down the system inside and out before starting this at the breakers. Let me know how that goes. That is a non communicating thermostat so it will be probably 4 wires. Red, brown, yellow and yellow with a blue stripe.
Not many hardware stores or home centers have capacitors. Most likely we need to find an HVaC supply house, a grainger. What is your zip code and I'll see what may be around. Disregard. Wrong page and I can't delete my response.
Let me know how the wiring check goes